VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.6s

Alternatives: 10

Speedup: 1.1×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift-+.f64 N/A

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

   2. *-lft-identity N/A

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

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

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

   4. lower--.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lift-/.f64 N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. metadata-eval N/A

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

   13. lift-/.f64 N/A

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

   14. div-inv N/A

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

   15. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} - \frac{-1}{\sin B}\\ \mathbf{if}\;t\_0 \leq -200 \lor \neg \left(t\_0 \leq 20\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ -1.0 (tan B))) (/ -1.0 (sin B)))))
   (if (or (<= t_0 -200.0) (not (<= t_0 20.0)))
     (/ (- 1.0 x) (tan B))
     (/ (- 1.0 x) (sin B)))))

C

double code(double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) - (-1.0 / sin(B));
	double tmp;
	if ((t_0 <= -200.0) || !(t_0 <= 20.0)) {
		tmp = (1.0 - x) / tan(B);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) - ((-1.0d0) / sin(b))
    if ((t_0 <= (-200.0d0)) .or. (.not. (t_0 <= 20.0d0))) then
        tmp = (1.0d0 - x) / tan(b)
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) - (-1.0 / Math.sin(B));
	double tmp;
	if ((t_0 <= -200.0) || !(t_0 <= 20.0)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (x * (-1.0 / math.tan(B))) - (-1.0 / math.sin(B))
	tmp = 0
	if (t_0 <= -200.0) or not (t_0 <= 20.0):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) - Float64(-1.0 / sin(B)))
	tmp = 0.0
	if ((t_0 <= -200.0) || !(t_0 <= 20.0))
		tmp = Float64(Float64(1.0 - x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (x * (-1.0 / tan(B))) - (-1.0 / sin(B));
	tmp = 0.0;
	if ((t_0 <= -200.0) || ~((t_0 <= 20.0)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	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 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -200.0], N[Not[LessEqual[t$95$0, 20.0]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -200 or 20 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   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. lift-+.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. lower--.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 98.8

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

   9. Applied rewrites 98.8%

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

   if -200 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 20

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. div-add-rev N/A

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

      9. lft-mult-inverse N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 96.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} - \frac{-1}{\sin B} \leq -200 \lor \neg \left(x \cdot \frac{-1}{\tan B} - \frac{-1}{\sin B} \leq 20\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 inf

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. associate-*r/ N/A

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

   5. div-add-rev N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r* N/A

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

   8. neg-mul-1 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right) \cdot B, B, \mathsf{fma}\left(0.022222222222222223, x, 0.019444444444444445\right)\right), B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 2.05)
   (/
    (fma
     (fma
      (fma
       (* (fma x 0.0021164021164021165 0.00205026455026455) B)
       B
       (fma 0.022222222222222223 x 0.019444444444444445))
      (* B B)
      (fma 0.3333333333333333 x 0.16666666666666666))
     (* B B)
     (- 1.0 x))
    B)
   (pow (sin B) -1.0)))

C

double code(double B, double x) {
	double tmp;
	if (B <= 2.05) {
		tmp = fma(fma(fma((fma(x, 0.0021164021164021165, 0.00205026455026455) * B), B, fma(0.022222222222222223, x, 0.019444444444444445)), (B * B), fma(0.3333333333333333, x, 0.16666666666666666)), (B * B), (1.0 - x)) / B;
	} else {
		tmp = pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 2.05)
		tmp = Float64(fma(fma(fma(Float64(fma(x, 0.0021164021164021165, 0.00205026455026455) * B), B, fma(0.022222222222222223, x, 0.019444444444444445)), Float64(B * B), fma(0.3333333333333333, x, 0.16666666666666666)), Float64(B * B), Float64(1.0 - x)) / B);
	else
		tmp = sin(B) ^ -1.0;
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 2.05], N[(N[(N[(N[(N[(N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision] * B), $MachinePrecision] * B + N[(0.022222222222222223 * x + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.0499999999999998

   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 67.1%

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

   if 2.0499999999999998 < B

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. div-add-rev N/A

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

      9. lft-mult-inverse N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-sin.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 41.9

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

   10. Applied rewrites 41.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right) \cdot B, B, \mathsf{fma}\left(0.022222222222222223, x, 0.019444444444444445\right)\right), B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift-+.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-/.f64 N/A

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

   6. un-div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

   10. *-rgt-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-*.f64 83.8

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

4. Applied rewrites 83.8%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-*l/ N/A

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

   8. div-add-rev N/A

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

   9. lft-mult-inverse N/A

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

   10. lower-/.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-sin.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in B around 0

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

10. Applied rewrites 72.2%

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

Alternative 6: 51.1% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. lift-+.f64 N/A

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

   2. *-lft-identity N/A

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

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

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

   4. lower--.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lift-/.f64 N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. metadata-eval N/A

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

   13. lift-/.f64 N/A

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

   14. div-inv N/A

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

   15. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. mul-1-neg N/A

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

   12. sub-neg N/A

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

   13. lower--.f64 49.9

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

7. Applied rewrites 49.9%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3} \cdot \left(B \cdot x\right), B, 1 - x\right)}{B} \]
9. Step-by-step derivation

10. Applied rewrites 50.1%

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

11. Final simplification 50.1%

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

Alternative 7: 49.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -410 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -410.0) (not (<= x 1.0))) (/ (- x) B) (/ (+ x 1.0) B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -410.0) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = (x + 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 <= (-410.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = -x / b
    else
        tmp = (x + 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -410.0) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -410.0) or not (x <= 1.0):
		tmp = -x / B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -410.0) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -410.0) || ~((x <= 1.0)))
		tmp = -x / B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -410.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -410 or 1 < x

   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. lift-+.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. lower--.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 47.8

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

   7. Applied rewrites 47.8%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 47.4%

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

   11. Taylor expanded in B around 0

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

   13. Applied rewrites 47.1%

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

   if -410 < 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 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 52.8

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

   5. Applied rewrites 52.8%

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

   7. Applied rewrites 52.8%

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

   9. Applied rewrites 52.4%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -410 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-14} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.4e-14) (not (<= x 1.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.4e-14) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.4d-14)) .or. (.not. (x <= 1.0d0))) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -3.4e-14) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.4e-14) or not (x <= 1.0):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3.4e-14) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -3.4e-14) || ~((x <= 1.0)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.4e-14], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000003e-14 or 1 < x

   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. lift-+.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. lower--.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 47.0

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

   7. Applied rewrites 47.0%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 46.5%

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

   11. Taylor expanded in B around 0

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

   13. Applied rewrites 46.2%

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

   if -3.40000000000000003e-14 < 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 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 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-14} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% 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 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 49.9

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

5. Applied rewrites 49.9%

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

6. Final simplification 49.9%

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

Alternative 10: 25.8% 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 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 49.9

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

5. Applied rewrites 49.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.3%

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

9. Final simplification 25.3%

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

Reproduce

herbie shell --seed 2024305 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))