VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 10.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\sin B} \cdot \mathsf{fma}\left(x, \cos B, -1\right) \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (* (/ -1.0 (sin B)) (fma x (cos B) -1.0)))

C

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

Julia

function code(B, x)
	return Float64(Float64(-1.0 / sin(B)) * fma(x, cos(B), -1.0))
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around inf

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

   1. associate-/l* N/A

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

   2. rgt-mult-inverse N/A

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

   3. associate-*r/ N/A

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

   4. associate-/r* N/A

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

   5. distribute-lft-out-- N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. neg-sub0 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} + \frac{1}{x \cdot \sin B}\right) \]

   9. associate-+l- N/A

      \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{\cos B}{\sin B} - \frac{1}{x \cdot \sin B}\right)\right)} \]

   10. neg-sub0 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \mathsf{fma}\left(x, \cos B, -1\right)} \]
8. Add Preprocessing

Alternative 3: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2200:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 13000000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -2200.0)
   (- (/ 1.0 B) (/ x (tan B)))
   (if (<= x 13000000000.0) (- (/ 1.0 (sin B)) (/ x B)) (/ (- x) (tan B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -2200.0) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (x <= 13000000000.0) {
		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 <= (-2200.0d0)) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (x <= 13000000000.0d0) 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 <= -2200.0) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (x <= 13000000000.0) {
		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 <= -2200.0:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif x <= 13000000000.0:
		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 <= -2200.0)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (x <= 13000000000.0)
		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 <= -2200.0)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (x <= 13000000000.0)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = -x / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -2200.0], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 13000000000.0], 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 < -2200

   1. Initial program 99.4%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if -2200 < x < 1.3e10

   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

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{1}{\sin B} \]
   4. Step-by-step derivation

      1. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 1.3e10 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. /-rgt-identity N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. /-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. /-rgt-identity N/A

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

      15. /-rgt-identity 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2200:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 13000000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -1.35e-6) t_0 (if (<= x 2.3e-8) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -1.35e-6) {
		tmp = t_0;
	} else if (x <= 2.3e-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 / b) - (x / tan(b))
    if (x <= (-1.35d-6)) then
        tmp = t_0
    else if (x <= 2.3d-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 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -1.35e-6) {
		tmp = t_0;
	} else if (x <= 2.3e-8) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -1.35e-6:
		tmp = t_0
	elif x <= 2.3e-8:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = t_0;
	elseif (x <= 2.3e-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 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -1.35e-6)
		tmp = t_0;
	elseif (x <= 2.3e-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 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-6], t$95$0, If[LessEqual[x, 2.3e-8], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.34999999999999999e-6 or 2.3000000000000001e-8 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if -1.34999999999999999e-6 < x < 2.3000000000000001e-8

   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. lower-/.f64 N/A

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

      2. lower-sin.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 5: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.32) t_0 (if (<= x 1.0) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1.32) {
		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 = -x / tan(b)
    if (x <= (-1.32d0)) 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 = -x / Math.tan(B);
	double tmp;
	if (x <= -1.32) {
		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 = -x / math.tan(B)
	tmp = 0
	if x <= -1.32:
		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(-x) / tan(B))
	tmp = 0.0
	if (x <= -1.32)
		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 = -x / tan(B);
	tmp = 0.0;
	if (x <= -1.32)
		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[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32], 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 < -1.32000000000000006 or 1 < x

   1. Initial program 99.5%

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. /-rgt-identity N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. /-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. /-rgt-identity N/A

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

      15. /-rgt-identity 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.7

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

   9. Applied rewrites 96.7%

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

   if -1.32000000000000006 < x < 1

   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. lower-/.f64 N/A

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

      2. lower-sin.f64 96.8

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

   5. Applied rewrites 96.8%

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

Alternative 6: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 5.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), 0.019444444444444445\right)\right), \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)\right), 1\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 5.8)
   (/
    (-
     (fma
      (* B B)
      (fma
       (* B B)
       (fma
        x
        0.022222222222222223
        (fma
         B
         (* B (fma x 0.0021164021164021165 0.00205026455026455))
         0.019444444444444445))
       (fma x 0.3333333333333333 0.16666666666666666))
      1.0)
     x)
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 5.8) {
		tmp = (fma((B * B), fma((B * B), fma(x, 0.022222222222222223, fma(B, (B * fma(x, 0.0021164021164021165, 0.00205026455026455)), 0.019444444444444445)), fma(x, 0.3333333333333333, 0.16666666666666666)), 1.0) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 5.8)
		tmp = Float64(Float64(fma(Float64(B * B), fma(Float64(B * B), fma(x, 0.022222222222222223, fma(B, Float64(B * fma(x, 0.0021164021164021165, 0.00205026455026455)), 0.019444444444444445)), fma(x, 0.3333333333333333, 0.16666666666666666)), 1.0) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 5.8], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] * N[(x * 0.022222222222222223 + N[(B * N[(B * N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision]), $MachinePrecision] + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] + N[(x * 0.3333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.79999999999999982

   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

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{31}{15120} + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), 0.019444444444444445\right)\right), \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)\right), 1\right) - x}{B}} \]

   if 5.79999999999999982 < B

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 62.1

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

   5. Applied rewrites 62.1%

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

Alternative 7: 50.7% accurate, 9.0× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -1.0) {
		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 = -x / b
    if (x <= (-1.0d0)) 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 = -x / B;
	double tmp;
	if (x <= -1.0) {
		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 = -x / B
	tmp = 0
	if x <= -1.0:
		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(Float64(-x) / B)
	tmp = 0.0
	if (x <= -1.0)
		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 = -x / B;
	tmp = 0.0;
	if (x <= -1.0)
		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[((-x) / B), $MachinePrecision]}, If[LessEqual[x, -1.0], 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 < -1 or 1 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.1%

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

   if -1 < x < 1

   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. lower-/.f64 N/A

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

      2. lower--.f64 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.9%

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

Alternative 8: 51.7% accurate, 9.0× 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. Add Preprocessing

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 51.9

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

5. Applied rewrites 51.9%

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

7. Applied rewrites 51.9%

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

Alternative 9: 51.7% accurate, 15.5× 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. lower-/.f64 N/A

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

   2. lower--.f64 51.9

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

5. Applied rewrites 51.9%

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

Alternative 10: 26.6% accurate, 19.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 51.9

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

5. Applied rewrites 51.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 26.9%

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

Alternative 11: 2.6% accurate, 21.2× speedup

Math

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

FPCore

(FPCore (B x) :precision binary64 (* x (* B 0.3333333333333333)))

C

double code(double B, double x) {
	return x * (B * 0.3333333333333333);
}

Fortran

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

Java

public static double code(double B, double x) {
	return x * (B * 0.3333333333333333);
}

Python

def code(B, x):
	return x * (B * 0.3333333333333333)

Julia

function code(B, x)
	return Float64(x * Float64(B * 0.3333333333333333))
end

MATLAB

function tmp = code(B, x)
	tmp = x * (B * 0.3333333333333333);
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) + \color{blue}{-1 \cdot x}}{B} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{\color{blue}{1 + \left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + -1 \cdot x\right)}}{B} \]

   4. +-commutative N/A

      \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)}}{B} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)}{B}} \]

7. Applied rewrites 51.8%

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

8. Taylor expanded in B around inf

   \[\leadsto B \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{3} \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 2.9%

    \[\leadsto B \cdot \color{blue}{\mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)} \]

11. Taylor expanded in x around inf

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

13. Applied rewrites 3.1%

    \[\leadsto x \cdot \left(B \cdot \color{blue}{0.3333333333333333}\right) \]
14. Add Preprocessing

Alternative 12: 3.1% accurate, 38.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) + \color{blue}{-1 \cdot x}}{B} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{\color{blue}{1 + \left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + -1 \cdot x\right)}}{B} \]

   4. +-commutative N/A

      \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)}}{B} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right)}{B}} \]

7. Applied rewrites 51.8%

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

8. Taylor expanded in B around inf

   \[\leadsto B \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{3} \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 2.9%

    \[\leadsto B \cdot \color{blue}{\mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto B \cdot \frac{1}{6} \]
12. Step-by-step derivation

13. Applied rewrites 3.0%

    \[\leadsto B \cdot 0.16666666666666666 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))