VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.2s

Alternatives: 15

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, 0.7× speedup

Math

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

FPCore

(FPCore (B x) :precision binary64 (/ (- (/ (tan B) (sin B)) x) (tan B)))

C

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

Fortran

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

Java

public static double code(double B, double x) {
	return ((Math.tan(B) / Math.sin(B)) - x) / Math.tan(B);
}

Python

def code(B, x):
	return ((math.tan(B) / math.sin(B)) - x) / math.tan(B)

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(N[Tan[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Tan[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. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. div-inv 99.7%

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

   4. frac-sub 89.8%

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

   5. associate-/r* 99.7%

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

   6. *-un-lft-identity 99.7%

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

   7. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. sub-neg 99.7%

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

   3. associate-/l* 99.7%

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

7. Applied egg-rr 99.7%

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

   1. sub-neg 99.7%

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

   2. *-inverses 99.7%

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

   3. /-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;\sin B - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 31000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -10.2)
   (- (sin B) (/ x (tan B)))
   (if (<= x 31000000.0) (- (/ 1.0 (sin B)) (/ x B)) (/ (- x) (tan B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -10.2) {
		tmp = sin(B) - (x / tan(B));
	} else if (x <= 31000000.0) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = -x / tan(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 <= (-10.2d0)) then
        tmp = sin(b) - (x / tan(b))
    else if (x <= 31000000.0d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = -x / tan(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -10.2) {
		tmp = Math.sin(B) - (x / Math.tan(B));
	} else if (x <= 31000000.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = -x / Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -10.2:
		tmp = math.sin(B) - (x / math.tan(B))
	elif x <= 31000000.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = -x / math.tan(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -10.2)
		tmp = Float64(sin(B) - Float64(x / tan(B)));
	elseif (x <= 31000000.0)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -10.2)
		tmp = sin(B) - (x / tan(B));
	elseif (x <= 31000000.0)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = -x / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -10.2], N[(N[Sin[B], $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 31000000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -10.199999999999999

   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. Step-by-step derivation

      1. add-exp-log 47.0%

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

      2. log-rec 47.0%

         \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{\tan B} \]

   5. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 1.9%

         \[\leadsto e^{\color{blue}{\sqrt{-\log \sin B} \cdot \sqrt{-\log \sin B}}} \]

      2. sqrt-unprod 1.9%

         \[\leadsto e^{\color{blue}{\sqrt{\left(-\log \sin B\right) \cdot \left(-\log \sin B\right)}}} \]

      3. sqr-neg 1.9%

         \[\leadsto e^{\sqrt{\color{blue}{\log \sin B \cdot \log \sin B}}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{\color{blue}{\sqrt{\log \sin B} \cdot \sqrt{\log \sin B}}} \]

      5. add-sqr-sqrt 1.2%

         \[\leadsto e^{\color{blue}{\log \sin B}} \]

      6. add-exp-log 2.9%

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

      7. expm1-log1p-u 2.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin B\right)\right)} \]

      8. expm1-udef 2.6%

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

   7. Applied egg-rr 98.0%

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

      1. expm1-def 2.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin B\right)\right)} \]

      2. expm1-log1p 2.9%

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

   9. Simplified 98.0%

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

   if -10.199999999999999 < x < 3.1e7

   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 96.2%

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

   if 3.1e7 < 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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. div-inv 99.6%

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

      4. frac-sub 96.5%

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

      5. associate-/r* 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;\sin B - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 31000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]

Alternative 3: 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/ 99.7%

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

   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. Final simplification 99.7%

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

Alternative 4: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 5500000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -10.2) (not (<= x 5500000.0)))
   (/ (- x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -10.2) || !(x <= 5500000.0)) {
		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 <= (-10.2d0)) .or. (.not. (x <= 5500000.0d0))) 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 <= -10.2) || !(x <= 5500000.0)) {
		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 <= -10.2) or not (x <= 5500000.0):
		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 <= -10.2) || !(x <= 5500000.0))
		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 <= -10.2) || ~((x <= 5500000.0)))
		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, -10.2], N[Not[LessEqual[x, 5500000.0]], $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 < -10.199999999999999 or 5.5e6 < 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. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. div-inv 99.7%

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

      4. frac-sub 97.3%

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

      5. associate-/r* 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. *-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around inf 98.6%

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

      1. neg-mul-1 98.6%

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

   8. Simplified 98.6%

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

   if -10.199999999999999 < x < 5.5e6

   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 96.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 5500000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.2) (not (<= x 1.05))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.2) || !(x <= 1.05)) {
		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 <= (-1.2d0)) .or. (.not. (x <= 1.05d0))) 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 <= -1.2) || !(x <= 1.05)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.2) or not (x <= 1.05):
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.2) || !(x <= 1.05))
		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 <= -1.2) || ~((x <= 1.05)))
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.2], N[Not[LessEqual[x, 1.05]], $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 < -1.19999999999999996 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. 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. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. div-inv 99.7%

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

      4. frac-sub 97.4%

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

      5. associate-/r* 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. *-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around inf 97.2%

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

      1. neg-mul-1 97.2%

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

   8. Simplified 97.2%

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

   if -1.19999999999999996 < 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. 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 95.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.00042 \lor \neg \left(B \leq 1.3 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -0.00042) (not (<= B 1.3e+14)))
   (/ 1.0 (sin B))
   (-
    (+ (/ 1.0 B) (* B (+ 0.16666666666666666 (* x 0.3333333333333333))))
    (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -0.00042) || !(B <= 1.3e+14)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 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 ((b <= (-0.00042d0)) .or. (.not. (b <= 1.3d+14))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = ((1.0d0 / b) + (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -0.00042) || !(B <= 1.3e+14)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -0.00042) or not (B <= 1.3e+14):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -0.00042) || !(B <= 1.3e+14))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -0.00042) || ~((B <= 1.3e+14)))
		tmp = 1.0 / sin(B);
	else
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -0.00042], N[Not[LessEqual[B, 1.3e+14]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -4.2000000000000002e-4 or 1.3e14 < B

   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 x around 0 47.8%

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

   if -4.2000000000000002e-4 < B < 1.3e14

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in B around 0 97.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.00042 \lor \neg \left(B \leq 1.3 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \end{array} \]

Alternative 7: 56.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.25 \cdot 10^{+30}:\\ \;\;\;\;\sin B\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\sin B\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B -1.25e+30)
   (sin B)
   (if (<= B 1.4e+15)
     (-
      (+ (/ 1.0 B) (* B (+ 0.16666666666666666 (* x 0.3333333333333333))))
      (/ x B))
     (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= -1.25e+30) {
		tmp = sin(B);
	} else if (B <= 1.4e+15) {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	} else {
		tmp = 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 <= (-1.25d+30)) then
        tmp = sin(b)
    else if (b <= 1.4d+15) then
        tmp = ((1.0d0 / b) + (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))) - (x / b)
    else
        tmp = sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= -1.25e+30) {
		tmp = Math.sin(B);
	} else if (B <= 1.4e+15) {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	} else {
		tmp = Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= -1.25e+30:
		tmp = math.sin(B)
	elif B <= 1.4e+15:
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B)
	else:
		tmp = math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= -1.25e+30)
		tmp = sin(B);
	elseif (B <= 1.4e+15)
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))) - Float64(x / B));
	else
		tmp = sin(B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= -1.25e+30)
		tmp = sin(B);
	elseif (B <= 1.4e+15)
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	else
		tmp = sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, -1.25e+30], N[Sin[B], $MachinePrecision], If[LessEqual[B, 1.4e+15], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[Sin[B], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < -1.25e30 or 1.4e15 < B

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-rgt-identity 99.5%

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

   3. Simplified 99.5%

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

      1. add-exp-log 47.8%

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

      2. log-rec 47.8%

         \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{\tan B} \]

   5. Applied egg-rr 47.8%

      \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{\tan B} \]

   6. Taylor expanded in x around 0 23.0%

      \[\leadsto \color{blue}{e^{-\log \sin B}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 23.0%

         \[\leadsto e^{\color{blue}{\sqrt{-\log \sin B} \cdot \sqrt{-\log \sin B}}} \]

      2. sqrt-unprod 23.0%

         \[\leadsto e^{\color{blue}{\sqrt{\left(-\log \sin B\right) \cdot \left(-\log \sin B\right)}}} \]

      3. sqr-neg 23.0%

         \[\leadsto e^{\sqrt{\color{blue}{\log \sin B \cdot \log \sin B}}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{\color{blue}{\sqrt{\log \sin B} \cdot \sqrt{\log \sin B}}} \]

      5. add-sqr-sqrt 5.3%

         \[\leadsto e^{\color{blue}{\log \sin B}} \]

      6. add-exp-log 11.4%

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

      7. expm1-log1p-u 11.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin B\right)\right)} \]

      8. expm1-udef 11.4%

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

   8. Applied egg-rr 11.4%

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

      1. expm1-def 11.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin B\right)\right)} \]

      2. expm1-log1p 11.4%

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

   10. Simplified 11.4%

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

   if -1.25e30 < B < 1.4e15

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in B around 0 94.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.25 \cdot 10^{+30}:\\ \;\;\;\;\sin B\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\sin B\\ \end{array} \]

Alternative 8: 52.1% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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/ 99.7%

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

   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 48.1%

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

5. Final simplification 48.1%

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

Alternative 9: 52.2% 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 48.1%

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

   1. +-commutative 48.1%

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

   2. mul-1-neg 48.1%

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

   3. sub-neg 48.1%

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

   4. associate--l+ 48.1%

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

   5. *-commutative 48.1%

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

   6. *-commutative 48.1%

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

   7. div-sub 48.1%

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

6. Simplified 48.1%

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

7. Final simplification 48.1%

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

Alternative 10: 52.3% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(0.3333333333333333 * N[(B * x), $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/ 99.7%

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

   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. Step-by-step derivation

   1. add-exp-log 48.4%

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

   2. log-rec 48.4%

      \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{\tan B} \]

5. Applied egg-rr 48.4%

   \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{\tan B} \]

6. Taylor expanded in B around 0 22.9%

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

   1. neg-mul-1 22.9%

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

   2. +-commutative 22.9%

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

   3. associate-+r+ 22.9%

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

   4. +-commutative 22.9%

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

   5. unsub-neg 22.9%

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

   6. exp-neg 22.9%

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

   7. rem-exp-log 47.9%

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

   8. div-sub 47.9%

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

   9. *-commutative 47.9%

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

   10. *-commutative 47.9%

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

8. Simplified 47.9%

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

9. Final simplification 47.9%

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

Alternative 11: 52.1% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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 70.9%

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

5. Taylor expanded in B around 0 48.1%

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

6. Final simplification 48.1%

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

Alternative 12: 50.8% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.3 \lor \neg \left(x \leq 1.4 \cdot 10^{-13}\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.3) (not (<= x 1.4e-13))) (- (/ x B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.3) || !(x <= 1.4e-13)) {
		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 <= (-0.3d0)) .or. (.not. (x <= 1.4d-13))) 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 <= -0.3) || !(x <= 1.4e-13)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.3) or not (x <= 1.4e-13):
		tmp = -(x / B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.3) || !(x <= 1.4e-13))
		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 <= -0.3) || ~((x <= 1.4e-13)))
		tmp = -(x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.3], N[Not[LessEqual[x, 1.4e-13]], $MachinePrecision]], (-N[(x / B), $MachinePrecision]), N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.299999999999999989 or 1.4000000000000001e-13 < x

   1. Initial program 99.6%

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

      1. 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 44.5%

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

      1. mul-1-neg 44.5%

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

      2. sub-neg 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in x around inf 44.2%

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

      1. neg-mul-1 44.2%

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

      2. distribute-neg-frac 44.2%

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

   9. Simplified 44.2%

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

   if -0.299999999999999989 < x < 1.4000000000000001e-13

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 51.4%

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

      1. mul-1-neg 51.4%

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

      2. sub-neg 51.4%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in x around 0 49.8%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.3 \lor \neg \left(x \leq 1.4 \cdot 10^{-13}\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 13: 51.9% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. 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 47.6%

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

   1. mul-1-neg 47.6%

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

   2. sub-neg 47.6%

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

6. Simplified 47.6%

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

   1. div-sub 47.6%

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

8. Applied egg-rr 47.6%

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

9. Final simplification 47.6%

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

Alternative 14: 51.9% 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 47.6%

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

   1. mul-1-neg 47.6%

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

   2. sub-neg 47.6%

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

6. Simplified 47.6%

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

7. Final simplification 47.6%

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

Alternative 15: 26.0% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 47.6%

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

   1. mul-1-neg 47.6%

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

   2. sub-neg 47.6%

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

6. Simplified 47.6%

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

7. Taylor expanded in x around 0 24.2%

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

8. Final simplification 24.2%

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

Reproduce

herbie shell --seed 2023178 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))