VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 8.9s

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: 99.7% 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
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. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. sub-div N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 99.8

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. 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. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -14800000:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -14800000.0)
   (/ (* (- x) (cos B)) (sin B))
   (if (<= x 1.3e-12)
     (- (/ 1.0 (sin B)) (/ x B))
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -14800000.0) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (x <= 1.3e-12) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-14800000.0d0)) then
        tmp = (-x * cos(b)) / sin(b)
    else if (x <= 1.3d-12) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -14800000.0) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (x <= 1.3e-12) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -14800000.0:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif x <= 1.3e-12:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -14800000.0)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (x <= 1.3e-12)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -14800000.0)
		tmp = (-x * cos(B)) / sin(B);
	elseif (x <= 1.3e-12)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -14800000.0], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-12], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.48e7

   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. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   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 99.2

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

   9. Applied rewrites 99.2%

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

   if -1.48e7 < x < 1.29999999999999991e-12

   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

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.29999999999999991e-12 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14800000:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -14800000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -14800000.0)
   (/ (- x) (tan B))
   (if (<= x 1.3e-12)
     (- (/ 1.0 (sin B)) (/ x B))
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -14800000.0) {
		tmp = -x / tan(B);
	} else if (x <= 1.3e-12) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-14800000.0d0)) then
        tmp = -x / tan(b)
    else if (x <= 1.3d-12) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -14800000.0) {
		tmp = -x / Math.tan(B);
	} else if (x <= 1.3e-12) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -14800000.0:
		tmp = -x / math.tan(B)
	elif x <= 1.3e-12:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -14800000.0)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (x <= 1.3e-12)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -14800000.0)
		tmp = -x / tan(B);
	elseif (x <= 1.3e-12)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -14800000.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-12], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.48e7

   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-neg.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 99.0

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

   7. Applied rewrites 99.0%

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

   9. Applied rewrites 99.0%

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

   if -1.48e7 < x < 1.29999999999999991e-12

   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

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.29999999999999991e-12 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14800000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -14800000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 380:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -14800000.0)
     t_0
     (if (<= x 380.0) (- (/ 1.0 (sin B)) (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -14800000.0) {
		tmp = t_0;
	} else if (x <= 380.0) {
		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 = -x / tan(b)
    if (x <= (-14800000.0d0)) then
        tmp = t_0
    else if (x <= 380.0d0) 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 = -x / Math.tan(B);
	double tmp;
	if (x <= -14800000.0) {
		tmp = t_0;
	} else if (x <= 380.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if x <= -14800000.0:
		tmp = t_0
	elif x <= 380.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -14800000.0)
		tmp = t_0;
	elseif (x <= 380.0)
		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 = -x / tan(B);
	tmp = 0.0;
	if (x <= -14800000.0)
		tmp = t_0;
	elseif (x <= 380.0)
		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[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -14800000.0], t$95$0, If[LessEqual[x, 380.0], 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 < -1.48e7 or 380 < 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-neg.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 98.7

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 98.8%

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

   if -1.48e7 < x < 380

   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

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14800000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 380:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -14800000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 380:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -14800000.0) t_0 (if (<= x 380.0) (/ (- 1.0 x) (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -14800000.0) {
		tmp = t_0;
	} else if (x <= 380.0) {
		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 = -x / tan(b)
    if (x <= (-14800000.0d0)) then
        tmp = t_0
    else if (x <= 380.0d0) 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 = -x / Math.tan(B);
	double tmp;
	if (x <= -14800000.0) {
		tmp = t_0;
	} else if (x <= 380.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if x <= -14800000.0:
		tmp = t_0
	elif x <= 380.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -14800000.0)
		tmp = t_0;
	elseif (x <= 380.0)
		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 = -x / tan(B);
	tmp = 0.0;
	if (x <= -14800000.0)
		tmp = t_0;
	elseif (x <= 380.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -14800000.0], t$95$0, If[LessEqual[x, 380.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.48e7 or 380 < 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-neg.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 98.7

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 98.8%

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

   if -1.48e7 < x < 380

   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. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. 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 98.4

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

   9. Applied rewrites 98.4%

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

Alternative 6: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.5) t_0 (if (<= x 1.0) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / tan(b)
    if (x <= (-1.5d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if x <= -1.5:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -x / tan(B);
	tmp = 0.0;
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1 < 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-neg.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 98.7

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 98.8%

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

   if -1.5 < 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 x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 7: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.165:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right) \cdot x, B \cdot B, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.165)
   (+
    (/ (fma 0.16666666666666666 (* B B) 1.0) B)
    (/
     (fma
      (* (fma 0.022222222222222223 (* B B) 0.3333333333333333) x)
      (* B B)
      (- x))
     B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.165) {
		tmp = (fma(0.16666666666666666, (B * B), 1.0) / B) + (fma((fma(0.022222222222222223, (B * B), 0.3333333333333333) * x), (B * B), -x) / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.165)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(B * B), 1.0) / B) + Float64(fma(Float64(fma(0.022222222222222223, Float64(B * B), 0.3333333333333333) * x), Float64(B * B), Float64(-x)) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.165], N[(N[(N[(0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] + N[(N[(N[(N[(0.022222222222222223 * N[(B * B), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.165000000000000008

   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

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

      1. lower-/.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 65.5%

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

   9. Taylor expanded in B around 0

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\frac{1}{45}, B \cdot B, \frac{1}{3}\right), B \cdot B, -x\right)}{B} + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 65.4

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

   11. Applied rewrites 65.4%

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

   if 0.165000000000000008 < B

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 47.5

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

   5. Applied rewrites 47.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.165:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right) \cdot x, B \cdot B, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -1.75e+28) 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 <= -1.75e+28) {
		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 <= (-1.75d+28)) 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 <= -1.75e+28) {
		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 <= -1.75e+28:
		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 <= -1.75e+28)
		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 <= -1.75e+28)
		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, -1.75e+28], 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 < -1.75e28 or 1 < 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 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.6%

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

   if -1.75e28 < 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 48.8

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.7%

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

Alternative 9: 51.3% 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 48.2

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

5. Applied rewrites 48.2%

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

Alternative 10: 26.9% 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 48.2

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

5. Applied rewrites 48.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.5%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))