VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 7.0s

Alternatives: 10

Speedup: 0.7×

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: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin B}^{-1}\\ t_1 := x \cdot \frac{-1}{\tan B} + t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+143} \lor \neg \left(t\_1 \leq 10^{+48}\right):\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (pow (sin B) -1.0)) (t_1 (+ (* x (/ -1.0 (tan B))) t_0)))
   (if (or (<= t_1 -2e+143) (not (<= t_1 1e+48)))
     (+ (/ (- x) (tan B)) (/ (fma 0.16666666666666666 (* B B) 1.0) B))
     (+ (- (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = pow(sin(B), -1.0);
	double t_1 = (x * (-1.0 / tan(B))) + t_0;
	double tmp;
	if ((t_1 <= -2e+143) || !(t_1 <= 1e+48)) {
		tmp = (-x / tan(B)) + (fma(0.16666666666666666, (B * B), 1.0) / B);
	} else {
		tmp = -(x / B) + t_0;
	}
	return tmp;
}

Julia

function code(B, x)
	t_0 = sin(B) ^ -1.0
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + t_0)
	tmp = 0.0
	if ((t_1 <= -2e+143) || !(t_1 <= 1e+48))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(fma(0.16666666666666666, Float64(B * B), 1.0) / B));
	else
		tmp = Float64(Float64(-Float64(x / B)) + t_0);
	end
	return tmp
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+143], N[Not[LessEqual[t$95$1, 1e+48]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + t$95$0), $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))) < -2e143 or 1.00000000000000004e48 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.7

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

   7. Applied rewrites 85.7%

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

   if -2e143 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 1.00000000000000004e48

   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

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

      1. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq -2 \cdot 10^{+143} \lor \neg \left(x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq 10^{+48}\right):\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[B], $MachinePrecision] * x + -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. Taylor expanded in B around inf

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

   1. rgt-mult-inverse N/A

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

   2. associate-*r/ N/A

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

   3. associate-/r* N/A

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

   4. associate-/l* N/A

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

   5. distribute-lft-out-- N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. neg-sub0 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} + \frac{1}{x \cdot \sin B}\right) \]

   9. associate-+l- N/A

      \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{\cos B}{\sin B} - \frac{1}{x \cdot \sin B}\right)\right)} \]

   10. neg-sub0 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. distribute-rgt-in N/A

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

7. Applied rewrites 99.6%

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

Alternative 4: 57.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.075:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333 \cdot B\right) \cdot x + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.075)
   (/ (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x) B)
   (+ (* (* 0.3333333333333333 B) x) (pow (sin B) -1.0))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.075) {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	} else {
		tmp = ((0.3333333333333333 * B) * x) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.075)
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 * B) * x) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.075], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(0.3333333333333333 * B), $MachinePrecision] * x), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0749999999999999972

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if 0.0749999999999999972 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 30.8%

      \[\leadsto \left(0.3333333333333333 \cdot B\right) \cdot \color{blue}{x} + \frac{1}{\sin B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.075:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333 \cdot B\right) \cdot x + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(-\frac{x}{B}\right) + {\sin B}^{-1} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[((-N[(x / B), $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. 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 75.2

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

5. Applied rewrites 75.2%

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

6. Final simplification 75.2%

   \[\leadsto \left(-\frac{x}{B}\right) + {\sin B}^{-1} \]
7. Add Preprocessing

Alternative 6: 51.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r/ N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-frac N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 62.9

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

5. Applied rewrites 62.9%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 53.7

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

8. Applied rewrites 53.7%

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

9. Taylor expanded in B around 0

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

11. Applied rewrites 54.2%

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

12. Final simplification 54.2%

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

Alternative 7: 51.6% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 53.8

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

5. Applied rewrites 53.8%

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

6. Final simplification 53.8%

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

Alternative 8: 50.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 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 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.1%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.4%

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

4. Final simplification 52.3%

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

Alternative 9: 51.4% 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 53.6

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

5. Applied rewrites 53.6%

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

6. Final simplification 53.6%

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

Alternative 10: 26.7% accurate, 19.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 53.6

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

5. Applied rewrites 53.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 26.1%

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

9. Final simplification 26.1%

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

Reproduce

herbie shell --seed 2024302 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))