VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 7.7s

Alternatives: 11

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 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := t\_0 + {\sin B}^{-1}\\ \mathbf{if}\;t\_1 \leq -10000000 \lor \neg \left(t\_1 \leq 50\right):\\ \;\;\;\;t\_0 + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) - \left(x + 1\right) \cdot \left(x \cdot x\right)}{\left(x + 1\right) \cdot \left(x + 1\right)}}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))) (t_1 (+ t_0 (pow (sin B) -1.0))))
   (if (or (<= t_1 -10000000.0) (not (<= t_1 50.0)))
     (+ t_0 (pow B -1.0))
     (/
      (/ (- (+ x 1.0) (* (+ x 1.0) (* x x))) (* (+ x 1.0) (+ x 1.0)))
      (sin B)))))

C

double code(double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = t_0 + pow(sin(B), -1.0);
	double tmp;
	if ((t_1 <= -10000000.0) || !(t_1 <= 50.0)) {
		tmp = t_0 + pow(B, -1.0);
	} else {
		tmp = (((x + 1.0) - ((x + 1.0) * (x * x))) / ((x + 1.0) * (x + 1.0))) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = t_0 + (sin(b) ** (-1.0d0))
    if ((t_1 <= (-10000000.0d0)) .or. (.not. (t_1 <= 50.0d0))) then
        tmp = t_0 + (b ** (-1.0d0))
    else
        tmp = (((x + 1.0d0) - ((x + 1.0d0) * (x * x))) / ((x + 1.0d0) * (x + 1.0d0))) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = t_0 + Math.pow(Math.sin(B), -1.0);
	double tmp;
	if ((t_1 <= -10000000.0) || !(t_1 <= 50.0)) {
		tmp = t_0 + Math.pow(B, -1.0);
	} else {
		tmp = (((x + 1.0) - ((x + 1.0) * (x * x))) / ((x + 1.0) * (x + 1.0))) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = t_0 + math.pow(math.sin(B), -1.0)
	tmp = 0
	if (t_1 <= -10000000.0) or not (t_1 <= 50.0):
		tmp = t_0 + math.pow(B, -1.0)
	else:
		tmp = (((x + 1.0) - ((x + 1.0) * (x * x))) / ((x + 1.0) * (x + 1.0))) / math.sin(B)
	return tmp

Julia

function code(B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(t_0 + (sin(B) ^ -1.0))
	tmp = 0.0
	if ((t_1 <= -10000000.0) || !(t_1 <= 50.0))
		tmp = Float64(t_0 + (B ^ -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) - Float64(Float64(x + 1.0) * Float64(x * x))) / Float64(Float64(x + 1.0) * Float64(x + 1.0))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = t_0 + (sin(B) ^ -1.0);
	tmp = 0.0;
	if ((t_1 <= -10000000.0) || ~((t_1 <= 50.0)))
		tmp = t_0 + (B ^ -1.0);
	else
		tmp = (((x + 1.0) - ((x + 1.0) * (x * x))) / ((x + 1.0) * (x + 1.0))) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -10000000.0], N[Not[LessEqual[t$95$1, 50.0]], $MachinePrecision]], N[(t$95$0 + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -1e7 or 50 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 98.9%

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

   if -1e7 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 50

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      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 99.5

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

   4. Applied rewrites 99.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]
   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. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/l* N/A

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

      18. lift-*.f64 N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 98.3

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

   9. Applied rewrites 98.3%

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

   11. Applied rewrites 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq -10000000 \lor \neg \left(x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq 50\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) - \left(x + 1\right) \cdot \left(x \cdot x\right)}{\left(x + 1\right) \cdot \left(x + 1\right)}}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := t\_0 + {\sin B}^{-1}\\ \mathbf{if}\;t\_1 \leq -10000000 \lor \neg \left(t\_1 \leq 50\right):\\ \;\;\;\;t\_0 + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))) (t_1 (+ t_0 (pow (sin B) -1.0))))
   (if (or (<= t_1 -10000000.0) (not (<= t_1 50.0)))
     (+ t_0 (pow B -1.0))
     (/ (- 1.0 x) (sin B)))))

C

double code(double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = t_0 + pow(sin(B), -1.0);
	double tmp;
	if ((t_1 <= -10000000.0) || !(t_1 <= 50.0)) {
		tmp = t_0 + 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = t_0 + (sin(b) ** (-1.0d0))
    if ((t_1 <= (-10000000.0d0)) .or. (.not. (t_1 <= 50.0d0))) then
        tmp = t_0 + (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 t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = t_0 + Math.pow(Math.sin(B), -1.0);
	double tmp;
	if ((t_1 <= -10000000.0) || !(t_1 <= 50.0)) {
		tmp = t_0 + Math.pow(B, -1.0);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = t_0 + math.pow(math.sin(B), -1.0)
	tmp = 0
	if (t_1 <= -10000000.0) or not (t_1 <= 50.0):
		tmp = t_0 + math.pow(B, -1.0)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(t_0 + (sin(B) ^ -1.0))
	tmp = 0.0
	if ((t_1 <= -10000000.0) || !(t_1 <= 50.0))
		tmp = Float64(t_0 + (B ^ -1.0));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = t_0 + (sin(B) ^ -1.0);
	tmp = 0.0;
	if ((t_1 <= -10000000.0) || ~((t_1 <= 50.0)))
		tmp = t_0 + (B ^ -1.0);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -10000000.0], N[Not[LessEqual[t$95$1, 50.0]], $MachinePrecision]], N[(t$95$0 + 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 (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -1e7 or 50 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 98.9%

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

   if -1e7 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 50

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      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 99.5

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

   4. Applied rewrites 99.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]
   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. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/l* N/A

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

      18. lift-*.f64 N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 98.3

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

   9. Applied rewrites 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq -10000000 \lor \neg \left(x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq 50\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \mathbf{if}\;t\_0 \leq -10000000 \lor \neg \left(t\_0 \leq 50\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) - \left(x + 1\right) \cdot \left(x \cdot x\right)}{\left(x + 1\right) \cdot \left(x + 1\right)}}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (pow (sin B) -1.0))))
   (if (or (<= t_0 -10000000.0) (not (<= t_0 50.0)))
     (+ (/ (- x) (tan B)) (pow B -1.0))
     (/
      (/ (- (+ x 1.0) (* (+ x 1.0) (* x x))) (* (+ x 1.0) (+ x 1.0)))
      (sin B)))))

C

double code(double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + pow(sin(B), -1.0);
	double tmp;
	if ((t_0 <= -10000000.0) || !(t_0 <= 50.0)) {
		tmp = (-x / tan(B)) + pow(B, -1.0);
	} else {
		tmp = (((x + 1.0) - ((x + 1.0) * (x * x))) / ((x + 1.0) * (x + 1.0))) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) + (sin(b) ** (-1.0d0))
    if ((t_0 <= (-10000000.0d0)) .or. (.not. (t_0 <= 50.0d0))) then
        tmp = (-x / tan(b)) + (b ** (-1.0d0))
    else
        tmp = (((x + 1.0d0) - ((x + 1.0d0) * (x * x))) / ((x + 1.0d0) * (x + 1.0d0))) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) + Math.pow(Math.sin(B), -1.0);
	double tmp;
	if ((t_0 <= -10000000.0) || !(t_0 <= 50.0)) {
		tmp = (-x / Math.tan(B)) + Math.pow(B, -1.0);
	} else {
		tmp = (((x + 1.0) - ((x + 1.0) * (x * x))) / ((x + 1.0) * (x + 1.0))) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + math.pow(math.sin(B), -1.0)
	tmp = 0
	if (t_0 <= -10000000.0) or not (t_0 <= 50.0):
		tmp = (-x / math.tan(B)) + math.pow(B, -1.0)
	else:
		tmp = (((x + 1.0) - ((x + 1.0) * (x * x))) / ((x + 1.0) * (x + 1.0))) / math.sin(B)
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + (sin(B) ^ -1.0))
	tmp = 0.0
	if ((t_0 <= -10000000.0) || !(t_0 <= 50.0))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + (B ^ -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) - Float64(Float64(x + 1.0) * Float64(x * x))) / Float64(Float64(x + 1.0) * Float64(x + 1.0))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (x * (-1.0 / tan(B))) + (sin(B) ^ -1.0);
	tmp = 0.0;
	if ((t_0 <= -10000000.0) || ~((t_0 <= 50.0)))
		tmp = (-x / tan(B)) + (B ^ -1.0);
	else
		tmp = (((x + 1.0) - ((x + 1.0) * (x * x))) / ((x + 1.0) * (x + 1.0))) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -10000000.0], N[Not[LessEqual[t$95$0, 50.0]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -1e7 or 50 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. metadata-eval N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. rem-log-exp N/A

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

      7. pow-to-exp N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 45.7

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{e^{\mathsf{neg}\left(\log B\right)}} \]
   8. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 99.0

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

   9. Applied rewrites 99.0%

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

   if -1e7 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 50

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      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 99.5

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

   4. Applied rewrites 99.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]
   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. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/l* N/A

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

      18. lift-*.f64 N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 98.3

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

   9. Applied rewrites 98.3%

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

   11. Applied rewrites 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq -10000000 \lor \neg \left(x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq 50\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) - \left(x + 1\right) \cdot \left(x \cdot x\right)}{\left(x + 1\right) \cdot \left(x + 1\right)}}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 99.8

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

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. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. clear-num N/A

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

   17. associate-/l* N/A

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

   18. lift-*.f64 N/A

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

   19. sub-div N/A

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

   20. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

Alternative 6: 87.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+85} \lor \neg \left(x \leq 8.5 \cdot 10^{+54}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + 0.16666666666666666 \cdot B\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -8.6e+85) (not (<= x 8.5e+54)))
   (+ (* x (/ -1.0 (tan B))) (* 0.16666666666666666 B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -8.6e+85) || !(x <= 8.5e+54)) {
		tmp = (x * (-1.0 / tan(B))) + (0.16666666666666666 * B);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-8.6d+85)) .or. (.not. (x <= 8.5d+54))) then
        tmp = (x * ((-1.0d0) / tan(b))) + (0.16666666666666666d0 * b)
    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 <= -8.6e+85) || !(x <= 8.5e+54)) {
		tmp = (x * (-1.0 / Math.tan(B))) + (0.16666666666666666 * B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -8.6e+85) or not (x <= 8.5e+54):
		tmp = (x * (-1.0 / math.tan(B))) + (0.16666666666666666 * B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -8.6e+85) || !(x <= 8.5e+54))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(0.16666666666666666 * B));
	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 <= -8.6e+85) || ~((x <= 8.5e+54)))
		tmp = (x * (-1.0 / tan(B))) + (0.16666666666666666 * B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -8.6e+85], N[Not[LessEqual[x, 8.5e+54]], $MachinePrecision]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5999999999999998e85 or 8.4999999999999995e54 < 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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 78.3%

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

   if -8.5999999999999998e85 < x < 8.4999999999999995e54

   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. 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. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/l* N/A

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

      18. lift-*.f64 N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

   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 92.4

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

   9. Applied rewrites 92.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+85} \lor \neg \left(x \leq 8.5 \cdot 10^{+54}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + 0.16666666666666666 \cdot B\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
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. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. clear-num N/A

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

   17. associate-/l* N/A

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

   18. lift-*.f64 N/A

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

   19. sub-div N/A

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

   20. lower-/.f64 N/A

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

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 76.1

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

9. Applied rewrites 76.1%

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

Alternative 8: 51.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right) \cdot B, B, \mathsf{fma}\left(0.022222222222222223, x, 0.019444444444444445\right)\right), B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (/
  (fma
   (fma
    (fma
     (* (fma x 0.0021164021164021165 0.00205026455026455) B)
     B
     (fma 0.022222222222222223 x 0.019444444444444445))
    (* B B)
    (fma 0.3333333333333333 x 0.16666666666666666))
   (* B B)
   (- 1.0 x))
  B))

C

double code(double B, double x) {
	return fma(fma(fma((fma(x, 0.0021164021164021165, 0.00205026455026455) * B), B, fma(0.022222222222222223, x, 0.019444444444444445)), (B * B), fma(0.3333333333333333, x, 0.16666666666666666)), (B * B), (1.0 - x)) / B;
}

Julia

function code(B, x)
	return Float64(fma(fma(fma(Float64(fma(x, 0.0021164021164021165, 0.00205026455026455) * B), B, fma(0.022222222222222223, x, 0.019444444444444445)), Float64(B * B), fma(0.3333333333333333, x, 0.16666666666666666)), Float64(B * B), Float64(1.0 - x)) / B)
end

Wolfram

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

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 99.8

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

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

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

7. Applied rewrites 51.4%

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

8. Final simplification 51.4%

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

Alternative 9: 51.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   2. inv-pow N/A

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

   3. metadata-eval N/A

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

   4. pow-to-exp N/A

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

   5. lower-exp.f64 N/A

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

   6. rem-log-exp N/A

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

   7. pow-to-exp N/A

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

   8. metadata-eval N/A

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

   9. inv-pow N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 48.0

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

6. Applied rewrites 48.0%

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

7. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{B \cdot \left(e^{\mathsf{neg}\left(\log B\right)} + \frac{1}{3} \cdot \left(B \cdot x\right)\right) - x}{B}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

9. Applied rewrites 51.3%

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

10. Final simplification 51.3%

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

Alternative 10: 51.6% 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 50.6

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

5. Applied rewrites 50.6%

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

Alternative 11: 26.3% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[((-x) / 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 50.6

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

5. Applied rewrites 50.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 26.8%

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

Reproduce

herbie shell --seed 2024315 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))