VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 10.3s

Alternatives: 12

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := t\_0 + x \cdot \frac{-1}{\tan B}\\ t_2 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_1 \leq -200000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1 (+ t_0 (* x (/ -1.0 (tan B)))))
        (t_2 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= t_1 -200000000.0) t_2 (if (<= t_1 500.0) t_0 t_2))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = t_0 + (x * (-1.0 / tan(B)));
	double t_2 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (t_1 <= -200000000.0) {
		tmp = t_2;
	} else if (t_1 <= 500.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = t_0 + (x * ((-1.0d0) / tan(b)))
    t_2 = (1.0d0 / b) - (x / tan(b))
    if (t_1 <= (-200000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 500.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = t_0 + (x * (-1.0 / Math.tan(B)));
	double t_2 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (t_1 <= -200000000.0) {
		tmp = t_2;
	} else if (t_1 <= 500.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = t_0 + (x * (-1.0 / math.tan(B)))
	t_2 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if t_1 <= -200000000.0:
		tmp = t_2
	elif t_1 <= 500.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(t_0 + Float64(x * Float64(-1.0 / tan(B))))
	t_2 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_1 <= -200000000.0)
		tmp = t_2;
	elseif (t_1 <= 500.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 1.0 / sin(B);
	t_1 = t_0 + (x * (-1.0 / tan(B)));
	t_2 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (t_1 <= -200000000.0)
		tmp = t_2;
	elseif (t_1 <= 500.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200000000.0], t$95$2, If[LessEqual[t$95$1, 500.0], t$95$0, t$95$2]]]]]

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))) < -2e8 or 500 < (+.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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -2e8 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 500

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq -200000000:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq 500:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around inf

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

7. Applied rewrites 99.7%

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

Alternative 4: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -1.6) t_0 (if (<= x 0.0048) (- (/ 1.0 (sin B)) (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -1.6) {
		tmp = t_0;
	} else if (x <= 0.0048) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-1.6d0)) then
        tmp = t_0
    else if (x <= 0.0048d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -1.6) {
		tmp = t_0;
	} else if (x <= 0.0048) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -1.6:
		tmp = t_0
	elif x <= 0.0048:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.6)
		tmp = t_0;
	elseif (x <= 0.0048)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -1.6)
		tmp = t_0;
	elseif (x <= 0.0048)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6], t$95$0, If[LessEqual[x, 0.0048], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 0.00479999999999999958 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if -1.6000000000000001 < x < 0.00479999999999999958

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.49:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.3333333333333333, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), \mathsf{fma}\left(x, 0.022222222222222223, 0.019444444444444445\right)\right), 0.16666666666666666\right)\right), 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.49)
   (/
    (fma
     (* B B)
     (fma
      x
      0.3333333333333333
      (fma
       (* B B)
       (fma
        B
        (* B (fma x 0.0021164021164021165 0.00205026455026455))
        (fma x 0.022222222222222223 0.019444444444444445))
       0.16666666666666666))
     (- 1.0 x))
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.49) {
		tmp = fma((B * B), fma(x, 0.3333333333333333, fma((B * B), fma(B, (B * fma(x, 0.0021164021164021165, 0.00205026455026455)), fma(x, 0.022222222222222223, 0.019444444444444445)), 0.16666666666666666)), (1.0 - x)) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.49)
		tmp = Float64(fma(Float64(B * B), fma(x, 0.3333333333333333, fma(Float64(B * B), fma(B, Float64(B * fma(x, 0.0021164021164021165, 0.00205026455026455)), fma(x, 0.022222222222222223, 0.019444444444444445)), 0.16666666666666666)), Float64(1.0 - x)) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.49], N[(N[(N[(B * B), $MachinePrecision] * N[(x * 0.3333333333333333 + N[(N[(B * B), $MachinePrecision] * N[(B * N[(B * N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision]), $MachinePrecision] + N[(x * 0.022222222222222223 + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.48999999999999999

   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

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

   5. Applied rewrites 69.4%

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

   if 0.48999999999999999 < 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 x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 46.6

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

   5. Applied rewrites 46.6%

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

Alternative 6: 51.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[(B * B), $MachinePrecision] * N[(x * 0.3333333333333333 + N[(N[(B * B), $MachinePrecision] * N[(B * N[(B * N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision]), $MachinePrecision] + N[(x * 0.022222222222222223 + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $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. Taylor expanded in B around 0

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

5. Applied rewrites 51.2%

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

Alternative 7: 51.5% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around 0

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

   1. sub-neg N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right), 1 - x\right)}{B}} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-+r- N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 51.2

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

9. Applied rewrites 51.2%

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

Alternative 8: 51.5% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[(N[(B * B), $MachinePrecision] * N[(x * 0.3333333333333333 + 0.16666666666666666), $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. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 51.2

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

5. Applied rewrites 51.2%

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

Alternative 9: 50.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-B}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ x (- B))))
   (if (<= x -2.4e-8) t_0 (if (<= x 1.0) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (x <= -2.4e-8) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -b
    if (x <= (-2.4d-8)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (x <= -2.4e-8) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = x / -B
	tmp = 0
	if x <= -2.4e-8:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(x / Float64(-B))
	tmp = 0.0
	if (x <= -2.4e-8)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = x / -B;
	tmp = 0.0;
	if (x <= -2.4e-8)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(x / (-B)), $MachinePrecision]}, If[LessEqual[x, -2.4e-8], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999998e-8 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 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.9

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

   8. Applied rewrites 48.9%

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

   if -2.39999999999999998e-8 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 51.2

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

   8. Applied rewrites 51.2%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 51.2

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

5. Applied rewrites 51.2%

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

Alternative 11: 26.4% 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 x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 51.6

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

5. Applied rewrites 51.6%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 27.5

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

8. Applied rewrites 27.5%

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

Alternative 12: 2.8% accurate, 19.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around 0

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

   1. sub-neg N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 51.2%

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

8. Taylor expanded in B around inf

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 2.8

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

10. Applied rewrites 2.8%

    \[\leadsto \color{blue}{B \cdot \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))