VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 7.7s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(B, x):
	return (1.0 / math.sin(B)) - (x / math.tan(B))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. cancel-sign-sub-inv 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-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: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+21}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 445000000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -2.05e+21)
   (/ (- x) (tan B))
   (if (<= x 445000000.0) (/ (- 1.0 x) (sin B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -2.05e+21) {
		tmp = -x / tan(B);
	} else if (x <= 445000000.0) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = (1.0 / B) - (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 <= (-2.05d+21)) then
        tmp = -x / tan(b)
    else if (x <= 445000000.0d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -2.05e+21) {
		tmp = -x / Math.tan(B);
	} else if (x <= 445000000.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -2.05e+21:
		tmp = -x / math.tan(B)
	elif x <= 445000000.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -2.05e+21)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (x <= 445000000.0)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -2.05e+21)
		tmp = -x / tan(B);
	elseif (x <= 445000000.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -2.05e+21], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 445000000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.05e21

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. associate-/l* 99.8%

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

      2. tan-quot 99.8%

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

      3. expm1-log1p-u 40.5%

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

      4. expm1-udef 40.5%

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

   4. Applied egg-rr 40.5%

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

      1. expm1-def 40.5%

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

      2. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

   if -2.05e21 < x < 4.45e8

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

      2. +-commutative 99.9%

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

      3. cancel-sign-sub-inv 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. inv-pow 99.8%

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

      2. add-sqr-sqrt 54.7%

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

      3. unpow-prod-down 54.5%

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

   5. Applied egg-rr 54.5%

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

      1. pow-sqr 54.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{\left(2 \cdot -1\right)}} - \frac{x}{\tan B} \]

      2. metadata-eval 54.7%

         \[\leadsto {\left(\sqrt{\sin B}\right)}^{\color{blue}{-2}} - \frac{x}{\tan B} \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{-2}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around inf 99.8%

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

      1. div-sub 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in B around 0 98.6%

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

   if 4.45e8 < 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%

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

      2. +-commutative 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r/ 99.7%

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

      7. *-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 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+21}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 445000000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+21} \lor \neg \left(x \leq 520000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.05e+21) (not (<= x 520000000.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.05e+21) || !(x <= 520000000.0)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.05d+21)) .or. (.not. (x <= 520000000.0d0))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -2.05e+21) || !(x <= 520000000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.05e+21) or not (x <= 520000000.0):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.05e+21) || !(x <= 520000000.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -2.05e+21) || ~((x <= 520000000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.05e+21], N[Not[LessEqual[x, 520000000.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05e21 or 5.2e8 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. associate-/l* 99.4%

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

      2. tan-quot 99.5%

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

      3. expm1-log1p-u 51.7%

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

      4. expm1-udef 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. expm1-def 51.7%

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

      2. expm1-log1p 99.5%

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

   6. Simplified 99.5%

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

   if -2.05e21 < x < 5.2e8

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

      2. +-commutative 99.9%

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

      3. cancel-sign-sub-inv 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. inv-pow 99.8%

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

      2. add-sqr-sqrt 54.7%

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

      3. unpow-prod-down 54.5%

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

   5. Applied egg-rr 54.5%

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

      1. pow-sqr 54.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{\left(2 \cdot -1\right)}} - \frac{x}{\tan B} \]

      2. metadata-eval 54.7%

         \[\leadsto {\left(\sqrt{\sin B}\right)}^{\color{blue}{-2}} - \frac{x}{\tan B} \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\sin B}\right)}^{-2}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around inf 99.8%

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

      1. div-sub 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in B around 0 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+21} \lor \neg \left(x \leq 520000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.35) || !(x <= 1.0)) {
		tmp = -x / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.35d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = -x / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.35) || !(x <= 1.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.35) or not (x <= 1.0):
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.35) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.35) || ~((x <= 1.0)))
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.35], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001 or 1 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. associate-/l* 97.8%

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

      2. tan-quot 97.9%

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

      3. expm1-log1p-u 52.0%

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

      4. expm1-udef 52.0%

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

   4. Applied egg-rr 52.0%

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

      1. expm1-def 52.0%

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

      2. expm1-log1p 97.9%

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

   6. Simplified 97.9%

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

   if -1.3500000000000001 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.4%

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

4. Final simplification 98.1%

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

Alternative 5: 62.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 6.8e-13) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 6.8e-13) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 6.8d-13) then
        tmp = (1.0d0 - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 6.8e-13) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 6.8e-13:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 6.8e-13)
		tmp = Float64(Float64(1.0 - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 6.8e-13)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 6.8e-13], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.80000000000000031e-13

   1. Initial program 99.8%

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

   2. Taylor expanded in B around 0 66.6%

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

   if 6.80000000000000031e-13 < B

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 55.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 50.9% 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 99.8%

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

   1. distribute-lft-neg-in 99.8%

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

   2. +-commutative 99.8%

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

   3. cancel-sign-sub-inv 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-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 72.5%

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

5. Taylor expanded in B around 0 50.9%

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

   1. associate--l+ 50.9%

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

   2. div-sub 50.9%

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

7. Simplified 50.9%

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

8. Final simplification 50.9%

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

Alternative 7: 50.7% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in B around 0 50.6%

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

   1. associate--l+ 50.6%

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

   2. *-commutative 50.6%

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

   3. div-sub 50.7%

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

4. Simplified 50.7%

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

5. Taylor expanded in x around 0 50.7%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot B} + \frac{1 - x}{B} \]
6. Step-by-step derivation

   1. *-commutative 50.7%

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

7. Simplified 50.7%

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

8. Final simplification 50.7%

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

Alternative 8: 50.6% 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 99.8%

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

2. Taylor expanded in B around 0 50.6%

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

3. Final simplification 50.6%

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

Alternative 9: 26.5% accurate, 52.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in B around 0 50.6%

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

3. Taylor expanded in x around inf 24.3%

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

   1. neg-mul-1 24.3%

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

   2. distribute-neg-frac 24.3%

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

5. Simplified 24.3%

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

6. Final simplification 24.3%

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

Reproduce

herbie shell --seed 2023338 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))