VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.9s

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 N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\color{blue}{x} \cdot \frac{1}{\tan B}\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(x \cdot \frac{1}{\tan B}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x \cdot 1}{\color{blue}{\tan B}}\right)\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\tan \color{blue}{B}}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{\tan B}\right)\right) \]

   9. tan-lowering-tan.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

3. Simplified 99.8%

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

Alternative 2: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -4.2e-8)
   (/ (- 1.0 x) (tan B))
   (if (<= x 5.3e-5) (/ 1.0 (sin B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -4.2e-8) {
		tmp = (1.0 - x) / tan(B);
	} else if (x <= 5.3e-5) {
		tmp = 1.0 / 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 <= (-4.2d-8)) then
        tmp = (1.0d0 - x) / tan(b)
    else if (x <= 5.3d-5) then
        tmp = 1.0d0 / 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 <= -4.2e-8) {
		tmp = (1.0 - x) / Math.tan(B);
	} else if (x <= 5.3e-5) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -4.2e-8:
		tmp = (1.0 - x) / math.tan(B)
	elif x <= 5.3e-5:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -4.2e-8)
		tmp = Float64(Float64(1.0 - x) / tan(B));
	elseif (x <= 5.3e-5)
		tmp = Float64(1.0 / 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 <= -4.2e-8)
		tmp = (1.0 - x) / tan(B);
	elseif (x <= 5.3e-5)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -4.2e-8], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.3e-5], N[(1.0 / 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 < -4.19999999999999989e-8

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in N/A

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

      2. un-div-inv N/A

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

      3. *-lft-identity N/A

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

      4. associate-/r* N/A

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

      5. /-rgt-identity N/A

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

      6. frac-add N/A

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

      7. *-rgt-identity N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(x\right)}{1} \cdot 1 + \tan B \cdot \frac{1}{\sin B}\right), \color{blue}{\tan B}\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\left(0 - x\right) + \frac{\tan B}{\sin B}}{\tan B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - x\right)}, \mathsf{tan.f64}\left(B\right)\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 96.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

   7. Simplified 96.6%

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

   if -4.19999999999999989e-8 < x < 5.3000000000000001e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 99.5%

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

   if 5.3000000000000001e-5 < 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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\color{blue}{x} \cdot \frac{1}{\tan B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x \cdot 1}{\color{blue}{\tan B}}\right)\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\tan \color{blue}{B}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{\tan B}\right)\right) \]

      9. tan-lowering-tan.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 98.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 98.9%

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

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) (tan B))))
   (if (<= x -8.2e-9) t_0 (if (<= x 3.7e-8) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / tan(B);
	double tmp;
	if (x <= -8.2e-9) {
		tmp = t_0;
	} else if (x <= 3.7e-8) {
		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) / tan(b)
    if (x <= (-8.2d-9)) then
        tmp = t_0
    else if (x <= 3.7d-8) 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) / Math.tan(B);
	double tmp;
	if (x <= -8.2e-9) {
		tmp = t_0;
	} else if (x <= 3.7e-8) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / math.tan(B)
	tmp = 0
	if x <= -8.2e-9:
		tmp = t_0
	elif x <= 3.7e-8:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 - x) / tan(B))
	tmp = 0.0
	if (x <= -8.2e-9)
		tmp = t_0;
	elseif (x <= 3.7e-8)
		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) / tan(B);
	tmp = 0.0;
	if (x <= -8.2e-9)
		tmp = t_0;
	elseif (x <= 3.7e-8)
		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] / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.2e-9], t$95$0, If[LessEqual[x, 3.7e-8], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2000000000000006e-9 or 3.7e-8 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in N/A

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

      2. un-div-inv N/A

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

      3. *-lft-identity N/A

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

      4. associate-/r* N/A

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

      5. /-rgt-identity N/A

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

      6. frac-add N/A

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

      7. *-rgt-identity N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(x\right)}{1} \cdot 1 + \tan B \cdot \frac{1}{\sin B}\right), \color{blue}{\tan B}\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\left(0 - x\right) + \frac{\tan B}{\sin B}}{\tan B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - x\right)}, \mathsf{tan.f64}\left(B\right)\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 97.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{tan.f64}\left(\color{blue}{B}\right)\right) \]

   7. Simplified 97.8%

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

   if -8.2000000000000006e-9 < x < 3.7e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 99.5%

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

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0 - x}{\tan B}\\ \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 0.0 x) (tan B))))
   (if (<= x -1.7) t_0 (if (<= x 1.0) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (0.0 - x) / tan(B);
	double tmp;
	if (x <= -1.7) {
		tmp = t_0;
	} else if (x <= 1.0) {
		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 = (0.0d0 - x) / tan(b)
    if (x <= (-1.7d0)) then
        tmp = t_0
    else if (x <= 1.0d0) 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 = (0.0 - x) / Math.tan(B);
	double tmp;
	if (x <= -1.7) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (0.0 - x) / math.tan(B)
	tmp = 0
	if x <= -1.7:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(0.0 - x) / tan(B))
	tmp = 0.0
	if (x <= -1.7)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (0.0 - x) / tan(B);
	tmp = 0.0;
	if (x <= -1.7)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(0.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996 or 1 < x

   1. Initial program 99.6%

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

      1. +-commutative N/A

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

      2. div-inv N/A

         \[\leadsto \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{\tan B}\right)\right) \]

      3. sub-neg N/A

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

      4. clear-num N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. frac-sub N/A

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

      8. associate-/r* N/A

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

      9. div-inv N/A

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

      10. clear-num N/A

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

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot \frac{\tan B}{x} - \left(\mathsf{neg}\left(\sin B\right)\right) \cdot 1}{\mathsf{neg}\left(\sin B\right)}\right), \color{blue}{\left(\frac{x}{\tan B}\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{-1}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 97.2%

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

   if -1.69999999999999996 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 99.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{0 - x}{\tan B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.036:\\ \;\;\;\;\frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223 + B \cdot \left(B \cdot \left(x \cdot 0.0021164021164021165\right)\right)\right) + x \cdot 0.3333333333333333\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.036)
   (/
    (+
     1.0
     (-
      (*
       (* B B)
       (+
        (*
         (* B B)
         (+
          (* x 0.022222222222222223)
          (* B (* B (* x 0.0021164021164021165)))))
        (* x 0.3333333333333333)))
      x))
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.036) {
		tmp = (1.0 + (((B * B) * (((B * B) * ((x * 0.022222222222222223) + (B * (B * (x * 0.0021164021164021165))))) + (x * 0.3333333333333333))) - 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 <= 0.036d0) then
        tmp = (1.0d0 + (((b * b) * (((b * b) * ((x * 0.022222222222222223d0) + (b * (b * (x * 0.0021164021164021165d0))))) + (x * 0.3333333333333333d0))) - 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 <= 0.036) {
		tmp = (1.0 + (((B * B) * (((B * B) * ((x * 0.022222222222222223) + (B * (B * (x * 0.0021164021164021165))))) + (x * 0.3333333333333333))) - x)) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.036:
		tmp = (1.0 + (((B * B) * (((B * B) * ((x * 0.022222222222222223) + (B * (B * (x * 0.0021164021164021165))))) + (x * 0.3333333333333333))) - x)) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.036)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(B * B) * Float64(Float64(Float64(B * B) * Float64(Float64(x * 0.022222222222222223) + Float64(B * Float64(B * Float64(x * 0.0021164021164021165))))) + Float64(x * 0.3333333333333333))) - 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 <= 0.036)
		tmp = (1.0 + (((B * B) * (((B * B) * ((x * 0.022222222222222223) + (B * (B * (x * 0.0021164021164021165))))) + (x * 0.3333333333333333))) - x)) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.036], N[(N[(1.0 + N[(N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(x * 0.022222222222222223), $MachinePrecision] + N[(B * N[(B * N[(x * 0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0359999999999999973

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\color{blue}{x} \cdot \frac{1}{\tan B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x \cdot 1}{\color{blue}{\tan B}}\right)\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\tan \color{blue}{B}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{\tan B}\right)\right) \]

      9. tan-lowering-tan.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   3. Simplified 99.8%

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

      1. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{1}{\tan B} \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\tan B}\right), \color{blue}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \tan B\right), x\right)\right) \]

      5. tan-lowering-tan.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right), x\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right), x\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 81.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)}, x\right)\right) \]

   9. Simplified 81.2%

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

   10. Taylor expanded in B around 0

       \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right) - x}{B}} \]

   12. Simplified 66.1%

       \[\leadsto \color{blue}{\frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223 + B \cdot \left(B \cdot \left(x \cdot 0.0021164021164021165\right)\right)\right) + x \cdot 0.3333333333333333\right) - x\right)}{B}} \]

   if 0.0359999999999999973 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 52.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 52.2%

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

Alternative 6: 49.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x}{B}\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x B))))
   (if (<= x -4.9e-11) t_0 (if (<= x 1.0) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = 0.0 - (x / B);
	double tmp;
	if (x <= -4.9e-11) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / 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 = 0.0d0 - (x / b)
    if (x <= (-4.9d-11)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = 0.0 - (x / B);
	double tmp;
	if (x <= -4.9e-11) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 0.0 - (x / B)
	tmp = 0
	if x <= -4.9e-11:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(0.0 - Float64(x / B))
	tmp = 0.0
	if (x <= -4.9e-11)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 0.0 - (x / B);
	tmp = 0.0;
	if (x <= -4.9e-11)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.9e-11], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8999999999999999e-11 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

   5. Simplified 56.9%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{B}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 56.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 56.3%

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

   if -4.8999999999999999e-11 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 49.2%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

   8. Simplified 49.2%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-11}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{B}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (if (<= x -5.1e+39) (/ B (* B B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if (x <= -5.1e+39) {
		tmp = B / (B * 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 <= (-5.1d+39)) then
        tmp = b / (b * 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 <= -5.1e+39) {
		tmp = B / (B * B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -5.1e+39:
		tmp = B / (B * B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -5.1e+39)
		tmp = Float64(B / Float64(B * B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -5.1e+39)
		tmp = B / (B * B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -5.1e+39], N[(B / N[(B * B), $MachinePrecision]), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0999999999999998e39

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 4.3%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

   8. Simplified 4.3%

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

      1. *-inverses N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(B, \color{blue}{\left(B \cdot B\right)}\right) \]

      4. *-lowering-*.f64 24.9%

         \[\leadsto \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(B, \color{blue}{B}\right)\right) \]

   10. Applied egg-rr 24.9%

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

   if -5.0999999999999998e39 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 60.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 60.8%

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

   6. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 29.9%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

   8. Simplified 29.9%

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

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 99.7%

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

3. Taylor expanded in B around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - x\right), \color{blue}{B}\right) \]

   2. --lowering--.f64 53.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

5. Simplified 53.3%

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

Alternative 9: 26.5% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

   2. sin-lowering-sin.f64 48.7%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

5. Simplified 48.7%

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

6. Taylor expanded in B around 0

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

   1. /-lowering-/.f64 24.4%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

8. Simplified 24.4%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))