VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 11.2s

Alternatives: 9

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, 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 egg-rr 99.8%

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

Alternative 2: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := t\_0 + x \cdot \frac{-1}{\tan B}\\ t_2 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -2e+24) t_2 (if (<= t_1 5000.0) 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 <= -2e+24) {
		tmp = t_2;
	} else if (t_1 <= 5000.0) {
		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 <= (-2d+24)) then
        tmp = t_2
    else if (t_1 <= 5000.0d0) 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 <= -2e+24) {
		tmp = t_2;
	} else if (t_1 <= 5000.0) {
		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 <= -2e+24:
		tmp = t_2
	elif t_1 <= 5000.0:
		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 <= -2e+24)
		tmp = t_2;
	elseif (t_1 <= 5000.0)
		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 <= -2e+24)
		tmp = t_2;
	elseif (t_1 <= 5000.0)
		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, -2e+24], t$95$2, If[LessEqual[t$95$1, 5000.0], 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))) < -2e24 or 5e3 < (+.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 egg-rr 99.9%

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

   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 99.9

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

   7. Simplified 99.9%

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

   if -2e24 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 5e3

   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 92.9

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

   5. Simplified 92.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq 5000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. unsub-neg N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. associate-*l/ N/A

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

   14. sub-div N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-*.f64 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 4: 62.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.3333333333333333, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), \mathsf{fma}\left(x, 0.022222222222222223, 0.019444444444444445\right)\right), 0.16666666666666666\right)\right), 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.4)
   (/
    (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 <= 0.4) {
		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 <= 0.4)
		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, 0.4], 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 < 0.40000000000000002

   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. Simplified 74.0%

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

   if 0.40000000000000002 < B

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 45.7

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

   5. Simplified 45.7%

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

Alternative 5: 49.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= x -1.2e-9) t_0 (if (<= x 1.0) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -1.2e-9) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (x <= (-1.2d-9)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -1.2e-9) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -(x / B)
	tmp = 0
	if x <= -1.2e-9:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (x <= -1.2e-9)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (x <= -1.2e-9)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[x, -1.2e-9], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-9 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 53.6

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

   5. Simplified 53.6%

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

   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 52.9

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

   8. Simplified 52.9%

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

   if -1.2e-9 < 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 97.3

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

   5. Simplified 97.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 55.3

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

   8. Simplified 55.3%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(B * N[(x * 0.3333333333333333), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / 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. 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 55.2

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

5. Simplified 55.2%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 55.1

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

8. Simplified 55.1%

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

9. Taylor expanded in x around 0

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

    1. sub-neg N/A

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

    2. distribute-lft-in N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. associate-+l+ N/A

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

    6. +-commutative N/A

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

    7. distribute-rgt-neg-out N/A

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

    8. associate-*r/ N/A

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

    9. *-rgt-identity N/A

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

    10. sub-neg N/A

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

    11. div-sub N/A

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

    12. sub-neg N/A

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

    13. mul-1-neg N/A

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

    14. *-commutative N/A

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

    15. associate-*r* N/A

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

    16. *-commutative N/A

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

    17. lower-fma.f64 N/A

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

    18. *-commutative N/A

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

    19. lower-*.f64 N/A

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

11. Simplified 55.4%

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

Alternative 7: 50.9% 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 55.2

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

5. Simplified 55.2%

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

Alternative 8: 26.3% 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 53.0

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

5. Simplified 53.0%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 30.1

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

8. Simplified 30.1%

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

Alternative 9: 3.2% accurate, 38.8× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = b * 0.16666666666666666d0
end function

Java

public static double code(double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(B, x):
	return B * 0.16666666666666666

Julia

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

MATLAB

function tmp = code(B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $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 55.2

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

5. Simplified 55.2%

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

6. Taylor expanded in B around inf

   \[\leadsto \color{blue}{B \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)} \]
7. 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 2.7

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

8. Simplified 2.7%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{B \cdot \frac{1}{6}} \]

    2. lower-*.f64 3.0

       \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

11. Simplified 3.0%

    \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))