VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 7.4s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3600000000000 \lor \neg \left(x \leq 3.2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3600000000000.0) (not (<= x 3.2e-8)))
   (+ (/ (- x) (tan B)) (pow B -1.0))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3600000000000.0) || !(x <= 3.2e-8)) {
		tmp = (-x / tan(B)) + pow(B, -1.0);
	} 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) :: tmp
    if ((x <= (-3600000000000.0d0)) .or. (.not. (x <= 3.2d-8))) then
        tmp = (-x / tan(b)) + (b ** (-1.0d0))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -3600000000000.0) || !(x <= 3.2e-8)) {
		tmp = (-x / Math.tan(B)) + Math.pow(B, -1.0);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3600000000000.0) or not (x <= 3.2e-8):
		tmp = (-x / math.tan(B)) + math.pow(B, -1.0)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3600000000000.0) || !(x <= 3.2e-8))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + (B ^ -1.0));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -3600000000000.0) || ~((x <= 3.2e-8)))
		tmp = (-x / tan(B)) + (B ^ -1.0);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3600000000000.0], N[Not[LessEqual[x, 3.2e-8]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e12 or 3.2000000000000002e-8 < x

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

   if -3.6e12 < x < 3.2000000000000002e-8

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. tan-quot N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3600000000000 \lor \neg \left(x \leq 3.2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3600000000000:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -3600000000000.0)
   (/ (* (- x) (cos B)) (sin B))
   (if (<= x 3.2e-8)
     (/ (- 1.0 x) (sin B))
     (+ (/ (- x) (tan B)) (pow B -1.0)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -3600000000000.0) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (x <= 3.2e-8) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = (-x / tan(B)) + pow(B, -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3600000000000.0d0)) then
        tmp = (-x * cos(b)) / sin(b)
    else if (x <= 3.2d-8) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = (-x / tan(b)) + (b ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -3600000000000.0) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (x <= 3.2e-8) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = (-x / Math.tan(B)) + Math.pow(B, -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -3600000000000.0:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif x <= 3.2e-8:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = (-x / math.tan(B)) + math.pow(B, -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -3600000000000.0)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (x <= 3.2e-8)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + (B ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -3600000000000.0)
		tmp = (-x * cos(B)) / sin(B);
	elseif (x <= 3.2e-8)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = (-x / tan(B)) + (B ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -3600000000000.0], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-8], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6e12

   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

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. tan-quot N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if -3.6e12 < x < 3.2000000000000002e-8

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. tan-quot N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 3.2000000000000002e-8 < x

   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

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3600000000000:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $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
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. associate-/r/ N/A

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

   12. associate-*l/ N/A

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

   13. sub-div N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 5: 51.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r/ N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-frac N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 67.1

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

5. Applied rewrites 67.1%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 52.5

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

8. Applied rewrites 52.5%

   \[\leadsto \mathsf{fma}\left(B \cdot x, 0.3333333333333333, \frac{-x}{B}\right) + \color{blue}{\frac{1}{B}} \]
9. Step-by-step derivation

10. Applied rewrites 52.5%

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

11. Final simplification 52.5%

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

Alternative 6: 51.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r/ N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-frac N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 67.1

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

5. Applied rewrites 67.1%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 52.5

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

8. Applied rewrites 52.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 52.5%

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

12. Final simplification 52.5%

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

Alternative 7: 76.8% 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
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. associate-/r/ N/A

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

   12. associate-*l/ N/A

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

   13. sub-div N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in B around 0

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

   1. lower--.f64 79.6

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

9. Applied rewrites 79.6%

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

Alternative 8: 50.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 220000\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 220000.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 220000.0)) {
		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 <= (-1.0d0)) .or. (.not. (x <= 220000.0d0))) 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 <= -1.0) || !(x <= 220000.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 220000.0):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 220000.0))
		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 <= -1.0) || ~((x <= 220000.0)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 220000.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 2.2e5 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 52.9%

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

   if -1 < x < 2.2e5

   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 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.3%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 220000\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 52.3

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

5. Applied rewrites 52.3%

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

Alternative 10: 26.5% 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 52.3

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

5. Applied rewrites 52.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))