VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 9.7s

Alternatives: 11

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: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in x around 0

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

   1. rgt-mult-inverse N/A

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*r/ N/A

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

   4. distribute-neg-frac N/A

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

   5. associate-/r* N/A

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

   6. distribute-rgt-neg-out N/A

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

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

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

   8. cancel-sign-sub-inv N/A

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

   9. cancel-sign-sub N/A

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

   10. +-commutative N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. *-commutative N/A

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

   14. lower--.f64 N/A

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

5. Simplified 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 3: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{if}\;x \leq -210:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))
   (if (<= x -210.0) t_0 (if (<= x 1.1) (- (/ 1.0 (sin B)) (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -210.0) {
		tmp = t_0;
	} else if (x <= 1.1) {
		tmp = (1.0 / sin(B)) - (x / 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 * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    if (x <= (-210.0d0)) then
        tmp = t_0
    else if (x <= 1.1d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -210.0) {
		tmp = t_0;
	} else if (x <= 1.1) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	tmp = 0
	if x <= -210.0:
		tmp = t_0
	elif x <= 1.1:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B))
	tmp = 0.0
	if (x <= -210.0)
		tmp = t_0;
	elseif (x <= 1.1)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (x * (-1.0 / tan(B))) + (1.0 / B);
	tmp = 0.0;
	if (x <= -210.0)
		tmp = t_0;
	elseif (x <= 1.1)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -210.0], t$95$0, If[LessEqual[x, 1.1], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -210 or 1.1000000000000001 < 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

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

      1. lower-/.f64 97.4

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

   5. Simplified 97.4%

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

   if -210 < x < 1.1000000000000001

   1. Initial program 99.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 97.6

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

   5. Simplified 97.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -210:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. 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 75.0

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

5. Simplified 75.0%

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

6. Final simplification 75.0%

   \[\leadsto \frac{1}{\sin B} - \frac{x}{B} \]
7. Add Preprocessing

Alternative 5: 62.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.235:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B, B \cdot \left(x \cdot 0.0021164021164021165\right), x \cdot 0.022222222222222223\right), x \cdot 0.3333333333333333\right), 1\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.235)
   (/
    (-
     (fma
      (* B B)
      (fma
       (* B B)
       (fma B (* B (* x 0.0021164021164021165)) (* x 0.022222222222222223))
       (* x 0.3333333333333333))
      1.0)
     x)
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.235) {
		tmp = (fma((B * B), fma((B * B), fma(B, (B * (x * 0.0021164021164021165)), (x * 0.022222222222222223)), (x * 0.3333333333333333)), 1.0) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.235)
		tmp = Float64(Float64(fma(Float64(B * B), fma(Float64(B * B), fma(B, Float64(B * Float64(x * 0.0021164021164021165)), Float64(x * 0.022222222222222223)), Float64(x * 0.3333333333333333)), 1.0) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.235], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] * N[(B * N[(B * N[(x * 0.0021164021164021165), $MachinePrecision]), $MachinePrecision] + N[(x * 0.022222222222222223), $MachinePrecision]), $MachinePrecision] + N[(x * 0.3333333333333333), $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 < 0.23499999999999999

   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

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

      1. lower-/.f64 78.7

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

   5. Simplified 78.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 66.5%

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

   if 0.23499999999999999 < B

   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 48.3

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

   5. Simplified 48.3%

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

Alternative 6: 50.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-/.f64 70.0

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

5. Simplified 70.0%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-lft-identity N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\left(B \cdot B\right)} \cdot \frac{1}{3} + -1, 1\right)}{B} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{B \cdot \left(B \cdot \frac{1}{3}\right)} + -1, 1\right)}{B} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(B, B \cdot \frac{1}{3}, -1\right)}, 1\right)}{B} \]

   14. lower-*.f64 47.5

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

8. Simplified 47.5%

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

Alternative 7: 48.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.66 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -1.0) t_0 (if (<= x 1.66e-13) (/ 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 <= 1.66e-13) {
		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 <= 1.66d-13) 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 <= 1.66e-13) {
		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 <= 1.66e-13:
		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 <= 1.66e-13)
		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 <= 1.66e-13)
		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, 1.66e-13], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.66e-13 < 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. lower-/.f64 N/A

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

      2. lower--.f64 49.0

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

   5. Simplified 49.0%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.7

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

   8. Simplified 47.7%

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

   if -1 < x < 1.66e-13

   1. Initial program 99.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 97.5

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

   5. Simplified 97.5%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 44.9

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

   8. Simplified 44.9%

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

Alternative 8: 50.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] - 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 x around 0

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

   1. rgt-mult-inverse N/A

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*r/ N/A

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

   4. distribute-neg-frac N/A

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

   5. associate-/r* N/A

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

   6. distribute-rgt-neg-out N/A

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

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

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

   8. cancel-sign-sub-inv N/A

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

   9. cancel-sign-sub N/A

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

   10. +-commutative N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. *-commutative N/A

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

   14. lower--.f64 N/A

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

5. Simplified 99.8%

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

6. Taylor expanded in B around 0

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

   1. div-sub N/A

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

   2. associate--l+ N/A

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

   3. distribute-rgt-out-- N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. div-sub N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 47.4

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

8. Simplified 47.4%

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

9. Taylor expanded in x around 0

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

11. Simplified 47.1%

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

Alternative 9: 50.1% 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 47.1

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

5. Simplified 47.1%

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

Alternative 10: 26.0% 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 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 52.5

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

5. Simplified 52.5%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 24.6

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

8. Simplified 24.6%

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

Alternative 11: 3.2% 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

3. Taylor expanded in x around 0

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

   1. rgt-mult-inverse N/A

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*r/ N/A

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

   4. distribute-neg-frac N/A

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

   5. associate-/r* N/A

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

   6. distribute-rgt-neg-out N/A

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

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

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

   8. cancel-sign-sub-inv N/A

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

   9. cancel-sign-sub N/A

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

   10. +-commutative N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. *-commutative N/A

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

   14. lower--.f64 N/A

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

5. Simplified 99.8%

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

6. Taylor expanded in B around 0

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

   1. div-sub N/A

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

   2. associate--l+ N/A

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

   3. distribute-rgt-out-- N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. div-sub N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 47.4

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

8. Simplified 47.4%

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

9. Taylor expanded in B around inf

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f64 3.0

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

11. Simplified 3.0%

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

12. Taylor expanded in x around 0

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{B \cdot \frac{1}{6}} \]

    2. lower-*.f64 3.2

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

14. Simplified 3.2%

    \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))