VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 10.7s

Alternatives: 12

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. clear-num N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. associate-/l/ N/A

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

   9. clear-num N/A

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

   10. sub-div N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 3: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -520:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \mathbf{elif}\;x \leq 0.25:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(B, \left(B \cdot B\right) \cdot -0.16666666666666666, B\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= x -520.0)
     (- (/ 1.0 B) t_0)
     (if (<= x 0.25)
       (/ (- 1.0 x) (sin B))
       (- (/ 1.0 (fma B (* (* B B) -0.16666666666666666) B)) t_0)))))

C

double code(double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (x <= -520.0) {
		tmp = (1.0 / B) - t_0;
	} else if (x <= 0.25) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = (1.0 / fma(B, ((B * B) * -0.16666666666666666), B)) - t_0;
	}
	return tmp;
}

Julia

function code(B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (x <= -520.0)
		tmp = Float64(Float64(1.0 / B) - t_0);
	elseif (x <= 0.25)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(1.0 / fma(B, Float64(Float64(B * B) * -0.16666666666666666), B)) - t_0);
	end
	return tmp
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520.0], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.25], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(B * N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -520

   1. Initial program 99.6%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if -520 < x < 0.25

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. clear-num N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-/l/ N/A

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

      9. clear-num N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if 0.25 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(B, \left(B \cdot B\right) \cdot -0.16666666666666666, B\right)}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -520:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -520.0) t_0 (if (<= x 2.0) (- (/ 1.0 (sin B)) (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -520.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		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 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-520.0d0)) then
        tmp = t_0
    else if (x <= 2.0d0) 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 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -520.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -520.0:
		tmp = t_0
	elif x <= 2.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -520.0)
		tmp = t_0;
	elseif (x <= 2.0)
		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 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -520.0)
		tmp = t_0;
	elseif (x <= 2.0)
		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[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520.0], t$95$0, If[LessEqual[x, 2.0], 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 < -520 or 2 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -520 < x < 2

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -520:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.02:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -520.0) t_0 (if (<= x 1.02) (/ (- 1.0 x) (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -520.0) {
		tmp = t_0;
	} else if (x <= 1.02) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-520.0d0)) then
        tmp = t_0
    else if (x <= 1.02d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -520.0) {
		tmp = t_0;
	} else if (x <= 1.02) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -520.0:
		tmp = t_0
	elif x <= 1.02:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -520.0)
		tmp = t_0;
	elseif (x <= 1.02)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -520.0)
		tmp = t_0;
	elseif (x <= 1.02)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520.0], t$95$0, If[LessEqual[x, 1.02], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -520 or 1.02 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -520 < x < 1.02

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. clear-num N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-/l/ N/A

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

      9. clear-num N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 98.9

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

   9. Applied rewrites 98.9%

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

Alternative 6: 63.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.28:\\ \;\;\;\;\frac{\left(\left(B \cdot B\right) \cdot \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), 0.019444444444444445\right)\right), \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)\right) - x\right) + 1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.28)
   (/
    (+
     (-
      (*
       (* B B)
       (fma
        B
        (*
         B
         (fma
          x
          0.022222222222222223
          (fma
           (* B B)
           (fma x 0.0021164021164021165 0.00205026455026455)
           0.019444444444444445)))
        (fma x 0.3333333333333333 0.16666666666666666)))
      x)
     1.0)
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.28) {
		tmp = ((((B * B) * fma(B, (B * fma(x, 0.022222222222222223, fma((B * B), fma(x, 0.0021164021164021165, 0.00205026455026455), 0.019444444444444445))), fma(x, 0.3333333333333333, 0.16666666666666666))) - x) + 1.0) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.28)
		tmp = Float64(Float64(Float64(Float64(Float64(B * B) * fma(B, Float64(B * fma(x, 0.022222222222222223, fma(Float64(B * B), fma(x, 0.0021164021164021165, 0.00205026455026455), 0.019444444444444445))), fma(x, 0.3333333333333333, 0.16666666666666666))) - x) + 1.0) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.28], N[(N[(N[(N[(N[(B * B), $MachinePrecision] * N[(B * N[(B * N[(x * 0.022222222222222223 + N[(N[(B * B), $MachinePrecision] * N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision] + 0.019444444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.3333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.28000000000000003

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(B \cdot B\right)} \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot \frac{1}{45} + \left(\left(B \cdot B\right) \cdot \left(x \cdot \frac{2}{945} + \frac{31}{15120}\right) + \frac{7}{360}\right)\right) + \left(x \cdot \frac{1}{3} + \frac{1}{6}\right)\right) + \left(1 - x\right)}{B} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot \left(x \cdot \frac{1}{45} + \left(\left(B \cdot B\right) \cdot \left(x \cdot \frac{2}{945} + \frac{31}{15120}\right) + \frac{7}{360}\right)\right) + \left(x \cdot \frac{1}{3} + \frac{1}{6}\right)\right) + \left(1 - x\right)}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot \frac{1}{45} + \left(\color{blue}{\left(B \cdot B\right)} \cdot \left(x \cdot \frac{2}{945} + \frac{31}{15120}\right) + \frac{7}{360}\right)\right) + \left(x \cdot \frac{1}{3} + \frac{1}{6}\right)\right) + \left(1 - x\right)}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot \frac{1}{45} + \left(\left(B \cdot B\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{2}{945}, \frac{31}{15120}\right)} + \frac{7}{360}\right)\right) + \left(x \cdot \frac{1}{3} + \frac{1}{6}\right)\right) + \left(1 - x\right)}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot \frac{1}{45} + \color{blue}{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{2}{945}, \frac{31}{15120}\right), \frac{7}{360}\right)}\right) + \left(x \cdot \frac{1}{3} + \frac{1}{6}\right)\right) + \left(1 - x\right)}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{45}, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{2}{945}, \frac{31}{15120}\right), \frac{7}{360}\right)\right)} + \left(x \cdot \frac{1}{3} + \frac{1}{6}\right)\right) + \left(1 - x\right)}{B} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \mathsf{fma}\left(x, \frac{1}{45}, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{2}{945}, \frac{31}{15120}\right), \frac{7}{360}\right)\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{3}, \frac{1}{6}\right)}\right) + \left(1 - x\right)}{B} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \color{blue}{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{1}{45}, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{2}{945}, \frac{31}{15120}\right), \frac{7}{360}\right)\right), \mathsf{fma}\left(x, \frac{1}{3}, \frac{1}{6}\right)\right)} + \left(1 - x\right)}{B} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{1}{45}, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{2}{945}, \frac{31}{15120}\right), \frac{7}{360}\right)\right), \mathsf{fma}\left(x, \frac{1}{3}, \frac{1}{6}\right)\right) + \color{blue}{\left(1 - x\right)}}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \left(B \cdot B\right) \cdot \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{1}{45}, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{2}{945}, \frac{31}{15120}\right), \frac{7}{360}\right)\right), \mathsf{fma}\left(x, \frac{1}{3}, \frac{1}{6}\right)\right)}}{B} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right)} + \left(B \cdot B\right) \cdot \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{1}{45}, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, \frac{2}{945}, \frac{31}{15120}\right), \frac{7}{360}\right)\right), \mathsf{fma}\left(x, \frac{1}{3}, \frac{1}{6}\right)\right)}{B} \]

   9. Applied rewrites 68.3%

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

   if 0.28000000000000003 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 56.1

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

   5. Applied rewrites 56.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.28:\\ \;\;\;\;\frac{\left(\left(B \cdot B\right) \cdot \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), 0.019444444444444445\right)\right), \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)\right) - x\right) + 1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 99.7%

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

4. Applied rewrites 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. clear-num N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. associate-/l/ N/A

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

   9. clear-num N/A

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

   10. sub-div N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in B around 0

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

   1. lower--.f64 75.8

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

9. Applied rewrites 75.8%

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

Alternative 8: 51.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right), 1\right) - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (/ (- (fma (* B B) (fma x 0.3333333333333333 0.16666666666666666) 1.0) x) B))

C

double code(double B, double x) {
	return (fma((B * B), fma(x, 0.3333333333333333, 0.16666666666666666), 1.0) - x) / B;
}

Julia

function code(B, x)
	return Float64(Float64(fma(Float64(B * B), fma(x, 0.3333333333333333, 0.16666666666666666), 1.0) - x) / B)
end

Wolfram

code[B_, x_] := N[(N[(N[(N[(B * B), $MachinePrecision] * N[(x * 0.3333333333333333 + 0.16666666666666666), $MachinePrecision] + 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 B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 52.8

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

5. Applied rewrites 52.8%

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

Alternative 9: 50.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 49.3

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

   5. Applied rewrites 49.3%

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

   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 46.8

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

   8. Applied rewrites 46.8%

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

   if -1 < 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 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 53.6

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

   8. Applied rewrites 53.6%

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

4. Final simplification 50.2%

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

Alternative 10: 51.5% accurate, 9.0× speedup

Math

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

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(B, Float64(B * 0.16666666666666666), 1.0) - x) / B)
end

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 73.6

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

5. Applied rewrites 73.6%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 52.8

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

8. Applied rewrites 52.8%

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

Alternative 11: 51.5% 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 52.7

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

5. Applied rewrites 52.7%

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

Alternative 12: 26.7% accurate, 19.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 52.7

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

5. Applied rewrites 52.7%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 27.8

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

8. Applied rewrites 27.8%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))