VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.3s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -380000000 \lor \neg \left(x \leq 3500000000\right):\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -380000000.0) (not (<= x 3500000000.0)))
   (/ (* (cos B) (- x)) (sin B))
   (+ (- (/ x B)) (pow (sin B) -1.0))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -380000000.0) || !(x <= 3500000000.0)) {
		tmp = (cos(B) * -x) / sin(B);
	} else {
		tmp = -(x / B) + pow(sin(B), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-380000000.0d0)) .or. (.not. (x <= 3500000000.0d0))) then
        tmp = (cos(b) * -x) / sin(b)
    else
        tmp = -(x / b) + (sin(b) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -380000000.0) || !(x <= 3500000000.0)) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else {
		tmp = -(x / B) + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -380000000.0) or not (x <= 3500000000.0):
		tmp = (math.cos(B) * -x) / math.sin(B)
	else:
		tmp = -(x / B) + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -380000000.0) || !(x <= 3500000000.0))
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	else
		tmp = Float64(Float64(-Float64(x / B)) + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -380000000.0) || ~((x <= 3500000000.0)))
		tmp = (cos(B) * -x) / sin(B);
	else
		tmp = -(x / B) + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -380000000.0], N[Not[LessEqual[x, 3500000000.0]], $MachinePrecision]], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e8 or 3.5e9 < x

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if -3.8e8 < x < 3.5e9

   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -380000000 \lor \neg \left(x \leq 3500000000\right):\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in B around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 4: 83.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+120} \lor \neg \left(x \leq 2.65 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\mathsf{fma}\left(0.16666666666666666 \cdot B, B, 1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.5e+120) (not (<= x 2.65e+19)))
   (+ (/ (- x) (tan B)) (/ (fma (* 0.16666666666666666 B) B 1.0) B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.5e+120) || !(x <= 2.65e+19)) {
		tmp = (-x / tan(B)) + (fma((0.16666666666666666 * B), B, 1.0) / B);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.5e+120) || !(x <= 2.65e+19))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(fma(Float64(0.16666666666666666 * B), B, 1.0) / B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.5e+120], N[Not[LessEqual[x, 2.65e+19]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.16666666666666666 * B), $MachinePrecision] * B + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e120 or 2.65e19 < x

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]
   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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if -1.5e120 < x < 2.65e19

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 93.4

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

   9. Applied rewrites 93.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+120} \lor \neg \left(x \leq 2.65 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\mathsf{fma}\left(0.16666666666666666 \cdot B, B, 1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[(0.3333333333333333 * B), $MachinePrecision] * x + 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{1}{\sin B} \]
6. 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. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-out N/A

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

   7. *-rgt-identity N/A

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

   8. associate-*r/ N/A

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

   9. rem-exp-log N/A

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

   10. exp-neg N/A

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

   11. distribute-rgt-out N/A

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

   12. mul-1-neg N/A

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

7. Applied rewrites 66.7%

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

8. Taylor expanded in B around 0

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

   1. lower-/.f64 51.0

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

10. Applied rewrites 51.0%

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

11. Final simplification 51.0%

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

Alternative 6: 51.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{1}{\sin B} \]
6. 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. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-out N/A

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

   7. *-rgt-identity N/A

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

   8. associate-*r/ N/A

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

   9. rem-exp-log N/A

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

   10. exp-neg N/A

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

   11. distribute-rgt-out N/A

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

   12. mul-1-neg N/A

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

7. Applied rewrites 66.7%

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

8. Taylor expanded in B around 0

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

   1. lower-/.f64 51.0

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

10. Applied rewrites 51.0%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 51.0%

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

14. Final simplification 51.0%

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

Alternative 7: 76.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in B around 0

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

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. lower--.f64 80.4

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

9. Applied rewrites 80.4%

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

Alternative 8: 51.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. clear-num N/A

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

   10. frac-sub N/A

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

   11. lower-/.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 87.9

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

4. Applied rewrites 87.9%

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

5. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 50.9%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 50.9%

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

11. Final simplification 50.9%

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

Alternative 9: 50.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.35e-6))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.35e-6)) {
		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 <= 1.35d-6))) 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 <= 1.35e-6)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.35e-6):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.35e-6))
		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 <= 1.35e-6)))
		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, 1.35e-6]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.34999999999999999e-6 < 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 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.6%

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

   if -1 < x < 1.34999999999999999e-6

   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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.7% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. *-commutative N/A

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

   5. log-pow N/A

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

   6. inv-pow N/A

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

   7. lift-/.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 47.7

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

4. Applied rewrites 47.7%

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

5. Taylor expanded in B around 0

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

7. Applied rewrites 50.6%

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

Alternative 11: 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 50.5

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

5. Applied rewrites 50.5%

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

6. Final simplification 50.5%

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

Alternative 12: 26.8% 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 50.5

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.7%

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

9. Final simplification 27.7%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))