VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 12.3s

Alternatives: 11

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. Step-by-step derivation

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.8%

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

   10. *-rgt-identity 99.8%

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

   11. tan-neg 99.8%

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

   12. distribute-neg-frac2 99.8%

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

   13. distribute-neg-frac 99.8%

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

   14. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-6} \lor \neg \left(x \leq 0.012\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.4e-6) (not (<= x 0.012)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.4e-6) || !(x <= 0.012)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.4d-6)) .or. (.not. (x <= 0.012d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.4e-6) || !(x <= 0.012)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.4e-6) or not (x <= 0.012):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.4e-6) || !(x <= 0.012))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.4e-6) || ~((x <= 0.012)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.4e-6], N[Not[LessEqual[x, 0.012]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999994e-6 or 0.012 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-frac-neg2 99.6%

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

      6. tan-neg 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 99.8%

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

   if -1.39999999999999994e-6 < x < 0.012

   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 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-6} \lor \neg \left(x \leq 0.012\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-6} \lor \neg \left(x \leq 0.00145\right):\\ \;\;\;\;x \cdot \left(\frac{\frac{1}{x} + -1}{B} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.2e-6) (not (<= x 0.00145)))
   (* x (+ (/ (+ (/ 1.0 x) -1.0) B) -1.0))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.2e-6) || !(x <= 0.00145)) {
		tmp = x * ((((1.0 / x) + -1.0) / B) + -1.0);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.2d-6)) .or. (.not. (x <= 0.00145d0))) then
        tmp = x * ((((1.0d0 / x) + (-1.0d0)) / b) + (-1.0d0))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.2e-6) || !(x <= 0.00145)) {
		tmp = x * ((((1.0 / x) + -1.0) / B) + -1.0);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.2e-6) or not (x <= 0.00145):
		tmp = x * ((((1.0 / x) + -1.0) / B) + -1.0)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.2e-6) || !(x <= 0.00145))
		tmp = Float64(x * Float64(Float64(Float64(Float64(1.0 / x) + -1.0) / B) + -1.0));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.2e-6) || ~((x <= 0.00145)))
		tmp = x * ((((1.0 / x) + -1.0) / B) + -1.0);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.2e-6], N[Not[LessEqual[x, 0.00145]], $MachinePrecision]], N[(x * N[(N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / B), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1999999999999999e-6 or 0.00145 < 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 49.0%

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

   4. Taylor expanded in x around inf 48.9%

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

      1. expm1-log1p-u 21.3%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{B \cdot x} - \frac{1}{B}\right)\right)} \]

      2. expm1-undefine 21.0%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} - \frac{1}{B}\right)} - 1\right)} \]

      3. sub-neg 21.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{B \cdot x} + \left(-\frac{1}{B}\right)}\right)} - 1\right) \]

      4. distribute-neg-frac 21.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \color{blue}{\frac{-1}{B}}\right)} - 1\right) \]

      5. metadata-eval 21.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{\color{blue}{-1}}{B}\right)} - 1\right) \]

   6. Applied egg-rr 21.0%

      \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{-1}{B}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. sub-neg 21.0%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{-1}{B}\right)} + \left(-1\right)\right)} \]

      2. log1p-undefine 21.0%

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

      3. rem-exp-log 48.6%

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

      4. metadata-eval 48.6%

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

      5. distribute-neg-frac 48.6%

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

      6. sub-neg 48.6%

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

      7. associate-/l/ 48.6%

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

      8. div-sub 48.6%

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

      9. sub-neg 48.6%

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

      10. metadata-eval 48.6%

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

      11. metadata-eval 48.6%

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

   8. Simplified 48.6%

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

   9. Taylor expanded in B around 0 51.0%

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

   if -1.1999999999999999e-6 < x < 0.00145

   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 98.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-6} \lor \neg \left(x \leq 0.00145\right):\\ \;\;\;\;x \cdot \left(\frac{\frac{1}{x} + -1}{B} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.1:\\ \;\;\;\;\frac{1 + \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.3333333333333333 + \left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223 + \left(B \cdot B\right) \cdot \left(\left(x \cdot 0.022222222222222223\right) \cdot -0.3333333333333333 + x \cdot 0.009523809523809525\right)\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1}{B \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 3.1)
   (/
    (+
     1.0
     (-
      (*
       (* B B)
       (+
        (* x 0.3333333333333333)
        (*
         (* B B)
         (+
          (* x 0.022222222222222223)
          (*
           (* B B)
           (+
            (* (* x 0.022222222222222223) -0.3333333333333333)
            (* x 0.009523809523809525)))))))
      x))
    B)
   (* x (+ -1.0 (/ 1.0 (* B x))))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 3.1) {
		tmp = (1.0 + (((B * B) * ((x * 0.3333333333333333) + ((B * B) * ((x * 0.022222222222222223) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + (x * 0.009523809523809525))))))) - x)) / B;
	} else {
		tmp = x * (-1.0 + (1.0 / (B * x)));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 3.1d0) then
        tmp = (1.0d0 + (((b * b) * ((x * 0.3333333333333333d0) + ((b * b) * ((x * 0.022222222222222223d0) + ((b * b) * (((x * 0.022222222222222223d0) * (-0.3333333333333333d0)) + (x * 0.009523809523809525d0))))))) - x)) / b
    else
        tmp = x * ((-1.0d0) + (1.0d0 / (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 3.1) {
		tmp = (1.0 + (((B * B) * ((x * 0.3333333333333333) + ((B * B) * ((x * 0.022222222222222223) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + (x * 0.009523809523809525))))))) - x)) / B;
	} else {
		tmp = x * (-1.0 + (1.0 / (B * x)));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 3.1:
		tmp = (1.0 + (((B * B) * ((x * 0.3333333333333333) + ((B * B) * ((x * 0.022222222222222223) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + (x * 0.009523809523809525))))))) - x)) / B
	else:
		tmp = x * (-1.0 + (1.0 / (B * x)))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 3.1)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(B * B) * Float64(Float64(x * 0.3333333333333333) + Float64(Float64(B * B) * Float64(Float64(x * 0.022222222222222223) + Float64(Float64(B * B) * Float64(Float64(Float64(x * 0.022222222222222223) * -0.3333333333333333) + Float64(x * 0.009523809523809525))))))) - x)) / B);
	else
		tmp = Float64(x * Float64(-1.0 + Float64(1.0 / Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 3.1)
		tmp = (1.0 + (((B * B) * ((x * 0.3333333333333333) + ((B * B) * ((x * 0.022222222222222223) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + (x * 0.009523809523809525))))))) - x)) / B;
	else
		tmp = x * (-1.0 + (1.0 / (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 3.1], N[(N[(1.0 + N[(N[(N[(B * B), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(N[(x * 0.022222222222222223), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(N[(N[(x * 0.022222222222222223), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(x * 0.009523809523809525), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(x * N[(-1.0 + N[(1.0 / N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.10000000000000009

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. tan-neg 99.8%

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

      7. cancel-sign-sub-inv 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. tan-neg 99.9%

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

      12. distribute-neg-frac2 99.9%

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

      13. distribute-neg-frac 99.9%

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

      14. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in B around 0 84.0%

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

   6. Taylor expanded in B around 0 66.1%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.3333333333333333 \cdot x + {B}^{2} \cdot \left(-0.1111111111111111 \cdot x + \left(0.13333333333333333 \cdot x + {B}^{2} \cdot \left(-0.3333333333333333 \cdot \left(-0.1111111111111111 \cdot x + 0.13333333333333333 \cdot x\right) + \left(-0.044444444444444446 \cdot x + 0.05396825396825397 \cdot x\right)\right)\right)\right)\right)\right) - x}{B}} \]
   7. Step-by-step derivation

   8. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{1 + \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.3333333333333333 + \left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223 + \left(B \cdot B\right) \cdot \left(-0.3333333333333333 \cdot \left(x \cdot 0.022222222222222223\right) + x \cdot 0.009523809523809525\right)\right)\right) - x\right)}{B}} \]

   if 3.10000000000000009 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 3.8%

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

   4. Taylor expanded in x around inf 3.8%

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

      1. expm1-log1p-u 3.3%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{B \cdot x} - \frac{1}{B}\right)\right)} \]

      2. expm1-undefine 3.0%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} - \frac{1}{B}\right)} - 1\right)} \]

      3. sub-neg 3.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{B \cdot x} + \left(-\frac{1}{B}\right)}\right)} - 1\right) \]

      4. distribute-neg-frac 3.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \color{blue}{\frac{-1}{B}}\right)} - 1\right) \]

      5. metadata-eval 3.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{\color{blue}{-1}}{B}\right)} - 1\right) \]

   6. Applied egg-rr 3.0%

      \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{-1}{B}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. sub-neg 3.0%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{-1}{B}\right)} + \left(-1\right)\right)} \]

      2. log1p-undefine 3.0%

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

      3. rem-exp-log 3.5%

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

      4. metadata-eval 3.5%

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

      5. distribute-neg-frac 3.5%

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

      6. sub-neg 3.5%

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

      7. associate-/l/ 3.5%

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

      8. div-sub 3.5%

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

      9. sub-neg 3.5%

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

      10. metadata-eval 3.5%

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

      11. metadata-eval 3.5%

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

   8. Simplified 3.5%

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

   9. Taylor expanded in x around 0 6.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1:\\ \;\;\;\;\frac{1 + \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.3333333333333333 + \left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223 + \left(B \cdot B\right) \cdot \left(\left(x \cdot 0.022222222222222223\right) \cdot -0.3333333333333333 + x \cdot 0.009523809523809525\right)\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1}{B \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.1:\\ \;\;\;\;\frac{1 + \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.3333333333333333 + \left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1}{B \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 3.1)
   (/
    (+
     1.0
     (-
      (*
       (* B B)
       (+ (* x 0.3333333333333333) (* (* B B) (* x 0.022222222222222223))))
      x))
    B)
   (* x (+ -1.0 (/ 1.0 (* B x))))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 3.1) {
		tmp = (1.0 + (((B * B) * ((x * 0.3333333333333333) + ((B * B) * (x * 0.022222222222222223)))) - x)) / B;
	} else {
		tmp = x * (-1.0 + (1.0 / (B * x)));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 3.1d0) then
        tmp = (1.0d0 + (((b * b) * ((x * 0.3333333333333333d0) + ((b * b) * (x * 0.022222222222222223d0)))) - x)) / b
    else
        tmp = x * ((-1.0d0) + (1.0d0 / (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 3.1) {
		tmp = (1.0 + (((B * B) * ((x * 0.3333333333333333) + ((B * B) * (x * 0.022222222222222223)))) - x)) / B;
	} else {
		tmp = x * (-1.0 + (1.0 / (B * x)));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 3.1:
		tmp = (1.0 + (((B * B) * ((x * 0.3333333333333333) + ((B * B) * (x * 0.022222222222222223)))) - x)) / B
	else:
		tmp = x * (-1.0 + (1.0 / (B * x)))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 3.1)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(B * B) * Float64(Float64(x * 0.3333333333333333) + Float64(Float64(B * B) * Float64(x * 0.022222222222222223)))) - x)) / B);
	else
		tmp = Float64(x * Float64(-1.0 + Float64(1.0 / Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 3.1)
		tmp = (1.0 + (((B * B) * ((x * 0.3333333333333333) + ((B * B) * (x * 0.022222222222222223)))) - x)) / B;
	else
		tmp = x * (-1.0 + (1.0 / (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 3.1], N[(N[(1.0 + N[(N[(N[(B * B), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(x * 0.022222222222222223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(x * N[(-1.0 + N[(1.0 / N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.10000000000000009

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. tan-neg 99.8%

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

      7. cancel-sign-sub-inv 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. tan-neg 99.9%

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

      12. distribute-neg-frac2 99.9%

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

      13. distribute-neg-frac 99.9%

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

      14. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in B around 0 84.0%

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

   6. Taylor expanded in B around 0 66.1%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.3333333333333333 \cdot x + {B}^{2} \cdot \left(-0.1111111111111111 \cdot x + 0.13333333333333333 \cdot x\right)\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. associate--l+ 66.1%

         \[\leadsto \frac{\color{blue}{1 + \left({B}^{2} \cdot \left(0.3333333333333333 \cdot x + {B}^{2} \cdot \left(-0.1111111111111111 \cdot x + 0.13333333333333333 \cdot x\right)\right) - x\right)}}{B} \]

      2. unpow2 66.1%

         \[\leadsto \frac{1 + \left(\color{blue}{\left(B \cdot B\right)} \cdot \left(0.3333333333333333 \cdot x + {B}^{2} \cdot \left(-0.1111111111111111 \cdot x + 0.13333333333333333 \cdot x\right)\right) - x\right)}{B} \]

      3. +-commutative 66.1%

         \[\leadsto \frac{1 + \left(\left(B \cdot B\right) \cdot \color{blue}{\left({B}^{2} \cdot \left(-0.1111111111111111 \cdot x + 0.13333333333333333 \cdot x\right) + 0.3333333333333333 \cdot x\right)} - x\right)}{B} \]

      4. unpow2 66.1%

         \[\leadsto \frac{1 + \left(\left(B \cdot B\right) \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot \left(-0.1111111111111111 \cdot x + 0.13333333333333333 \cdot x\right) + 0.3333333333333333 \cdot x\right) - x\right)}{B} \]

      5. distribute-rgt-out 66.1%

         \[\leadsto \frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \color{blue}{\left(x \cdot \left(-0.1111111111111111 + 0.13333333333333333\right)\right)} + 0.3333333333333333 \cdot x\right) - x\right)}{B} \]

      6. metadata-eval 66.1%

         \[\leadsto \frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot \color{blue}{0.022222222222222223}\right) + 0.3333333333333333 \cdot x\right) - x\right)}{B} \]

      7. *-commutative 66.1%

         \[\leadsto \frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223\right) + \color{blue}{x \cdot 0.3333333333333333}\right) - x\right)}{B} \]

   8. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223\right) + x \cdot 0.3333333333333333\right) - x\right)}{B}} \]

   if 3.10000000000000009 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 3.8%

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

   4. Taylor expanded in x around inf 3.8%

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

      1. expm1-log1p-u 3.3%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{B \cdot x} - \frac{1}{B}\right)\right)} \]

      2. expm1-undefine 3.0%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} - \frac{1}{B}\right)} - 1\right)} \]

      3. sub-neg 3.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{B \cdot x} + \left(-\frac{1}{B}\right)}\right)} - 1\right) \]

      4. distribute-neg-frac 3.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \color{blue}{\frac{-1}{B}}\right)} - 1\right) \]

      5. metadata-eval 3.0%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{\color{blue}{-1}}{B}\right)} - 1\right) \]

   6. Applied egg-rr 3.0%

      \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{-1}{B}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. sub-neg 3.0%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{B \cdot x} + \frac{-1}{B}\right)} + \left(-1\right)\right)} \]

      2. log1p-undefine 3.0%

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

      3. rem-exp-log 3.5%

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

      4. metadata-eval 3.5%

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

      5. distribute-neg-frac 3.5%

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

      6. sub-neg 3.5%

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

      7. associate-/l/ 3.5%

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

      8. div-sub 3.5%

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

      9. sub-neg 3.5%

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

      10. metadata-eval 3.5%

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

      11. metadata-eval 3.5%

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

   8. Simplified 3.5%

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

   9. Taylor expanded in x around 0 6.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1:\\ \;\;\;\;\frac{1 + \left(\left(B \cdot B\right) \cdot \left(x \cdot 0.3333333333333333 + \left(B \cdot B\right) \cdot \left(x \cdot 0.022222222222222223\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1}{B \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+25} \lor \neg \left(x \leq 6.8 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1e+25) (not (<= x 6.8e-15))) (/ x (- B)) (/ (+ 1.0 x) B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1e+25) || !(x <= 6.8e-15)) {
		tmp = x / -B;
	} else {
		tmp = (1.0 + x) / 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 <= (-1d+25)) .or. (.not. (x <= 6.8d-15))) then
        tmp = x / -b
    else
        tmp = (1.0d0 + x) / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1e+25) || !(x <= 6.8e-15)) {
		tmp = x / -B;
	} else {
		tmp = (1.0 + x) / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1e+25) or not (x <= 6.8e-15):
		tmp = x / -B
	else:
		tmp = (1.0 + x) / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1e+25) || !(x <= 6.8e-15))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 + x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1e+25) || ~((x <= 6.8e-15)))
		tmp = x / -B;
	else
		tmp = (1.0 + x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1e+25], N[Not[LessEqual[x, 6.8e-15]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000009e25 or 6.8000000000000001e-15 < 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 47.3%

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

   4. Taylor expanded in x around inf 47.4%

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

   if -1.00000000000000009e25 < x < 6.8000000000000001e-15

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.0%

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

   4. Applied egg-rr 96.4%

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

   5. Taylor expanded in B around 0 49.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+25} \lor \neg \left(x \leq 6.8 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+25} \lor \neg \left(x \leq 6.8 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1e+25) (not (<= x 6.8e-15))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1e+25) || !(x <= 6.8e-15)) {
		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 <= (-1d+25)) .or. (.not. (x <= 6.8d-15))) 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 <= -1e+25) || !(x <= 6.8e-15)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1e+25) or not (x <= 6.8e-15):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1e+25) || !(x <= 6.8e-15))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1e+25) || ~((x <= 6.8e-15)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1e+25], N[Not[LessEqual[x, 6.8e-15]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000009e25 or 6.8000000000000001e-15 < 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 47.3%

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

   4. Taylor expanded in x around inf 47.4%

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

   if -1.00000000000000009e25 < x < 6.8000000000000001e-15

   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 50.2%

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

   4. Taylor expanded in x around 0 49.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+25} \lor \neg \left(x \leq 6.8 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.6% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 / b) + (x * ((b * 0.3333333333333333d0) + ((-1.0d0) / b)))
end function

Java

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

Python

def code(B, x):
	return (1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / B)))

Julia

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

MATLAB

function tmp = code(B, x)
	tmp = (1.0 / B) + (x * ((B * 0.3333333333333333) + (-1.0 / B)));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.8%

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

   10. *-rgt-identity 99.8%

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

   11. tan-neg 99.8%

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

   12. distribute-neg-frac2 99.8%

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

   13. distribute-neg-frac 99.8%

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

   14. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in B around 0 71.8%

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

   1. clear-num 71.6%

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

   2. frac-sub 61.1%

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

   3. *-un-lft-identity 61.1%

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

   4. metadata-eval 61.1%

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

   5. div-inv 61.1%

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

   6. /-rgt-identity 61.1%

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

7. Applied egg-rr 61.1%

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

8. Taylor expanded in B around 0 48.9%

   \[\leadsto \color{blue}{\frac{x \cdot \left(\frac{1}{x} - 1\right) + {B}^{2} \cdot \left(0.3333333333333333 - 0.3333333333333333 \cdot \left(x \cdot \left(\frac{1}{x} - 1\right)\right)\right)}{B}} \]
9. Step-by-step derivation

   1. sub-neg 48.9%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)} + {B}^{2} \cdot \left(0.3333333333333333 - 0.3333333333333333 \cdot \left(x \cdot \left(\frac{1}{x} - 1\right)\right)\right)}{B} \]

   2. metadata-eval 48.9%

      \[\leadsto \frac{x \cdot \left(\frac{1}{x} + \color{blue}{-1}\right) + {B}^{2} \cdot \left(0.3333333333333333 - 0.3333333333333333 \cdot \left(x \cdot \left(\frac{1}{x} - 1\right)\right)\right)}{B} \]

   3. unpow2 48.9%

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

   4. sub-neg 48.9%

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

   5. *-commutative 48.9%

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

   6. distribute-rgt-neg-in 48.9%

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

   7. sub-neg 48.9%

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

   8. metadata-eval 48.9%

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

   9. metadata-eval 48.9%

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

10. Simplified 48.9%

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

11. Taylor expanded in x around 0 49.4%

    \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot B - \frac{1}{B}\right) + \frac{1}{B}} \]
12. Step-by-step derivation

    1. +-commutative 49.4%

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

    2. sub-neg 49.4%

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

    3. *-commutative 49.4%

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

    4. distribute-neg-frac 49.4%

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

    5. metadata-eval 49.4%

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

13. Simplified 49.4%

    \[\leadsto \color{blue}{\frac{1}{B} + x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)} \]
14. Add Preprocessing

Alternative 9: 52.4% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(1.0 - N[(N[(B * x), $MachinePrecision] / B), $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 48.9%

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

   1. div-sub 48.9%

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

   2. frac-sub 34.6%

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

   3. *-un-lft-identity 34.6%

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

5. Applied egg-rr 34.6%

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

   1. associate-/r* 49.0%

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

   2. div-sub 49.0%

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

   3. *-inverses 49.0%

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

7. Simplified 49.0%

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

Alternative 10: 52.3% accurate, 42.0× 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 48.9%

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

Alternative 11: 26.9% accurate, 70.0× 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 48.9%

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

4. Taylor expanded in x around 0 28.5%

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

Reproduce

herbie shell --seed 2024098 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))