VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.7s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. B = 5.32171492014721e+223
  2. x = -2.4880379775787554e+283

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $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. cancel-sign-sub-inv 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 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. Step-by-step derivation

   1. inv-pow 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00021 \lor \neg \left(x \leq 1.12\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sin B}{1 - x}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.00021) (not (<= x 1.12)))
   (- (/ 1.0 B) (/ x (tan B)))
   (pow (/ (sin B) (- 1.0 x)) -1.0)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.00021) || !(x <= 1.12)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = pow((sin(B) / (1.0 - x)), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.00021d0)) .or. (.not. (x <= 1.12d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (sin(b) / (1.0d0 - x)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -0.00021) || !(x <= 1.12)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = Math.pow((Math.sin(B) / (1.0 - x)), -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.00021) or not (x <= 1.12):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = math.pow((math.sin(B) / (1.0 - x)), -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.00021) || !(x <= 1.12))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(sin(B) / Float64(1.0 - x)) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.00021) || ~((x <= 1.12)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (sin(B) / (1.0 - x)) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.00021], N[Not[LessEqual[x, 1.12]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sin[B], $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1000000000000001e-4 or 1.1200000000000001 < 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. cancel-sign-sub-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Taylor expanded in B around 0 99.4%

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

   if -2.1000000000000001e-4 < x < 1.1200000000000001

   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. cancel-sign-sub-inv 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Step-by-step derivation

      1. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. inv-pow 99.8%

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

      2. frac-sub 84.0%

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

      3. *-un-lft-identity 84.0%

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

      4. *-commutative 84.0%

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

      5. *-commutative 84.0%

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

      6. div-sub 82.9%

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

      7. sub-neg 82.9%

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

      8. *-commutative 82.9%

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

      9. *-commutative 82.9%

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

      10. times-frac 84.0%

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

   7. Applied egg-rr 84.0%

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

      1. sub-neg 84.0%

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

      2. times-frac 82.9%

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

      3. *-commutative 82.9%

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

      4. div-sub 84.0%

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

      5. associate-/l/ 99.8%

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

      6. div-sub 99.8%

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

      7. *-inverses 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in B around 0 98.6%

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

       1. clear-num 98.6%

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

       2. inv-pow 98.6%

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

   12. Applied egg-rr 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00021 \lor \neg \left(x \leq 1.12\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sin B}{1 - x}\right)}^{-1}\\ \end{array} \]

Alternative 3: 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. cancel-sign-sub-inv 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 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. Final simplification 99.8%

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

Alternative 4: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00021 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.00021) (not (<= x 1.0)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.00021) || !(x <= 1.0)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 - x) / 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 <= (-0.00021d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -0.00021) || !(x <= 1.0)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.00021) or not (x <= 1.0):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.00021) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.00021) || ~((x <= 1.0)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.00021], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1000000000000001e-4 or 1 < 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. cancel-sign-sub-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Taylor expanded in B around 0 99.4%

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

   if -2.1000000000000001e-4 < x < 1

   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. cancel-sign-sub-inv 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Step-by-step derivation

      1. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. inv-pow 99.8%

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

      2. frac-sub 84.0%

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

      3. *-un-lft-identity 84.0%

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

      4. *-commutative 84.0%

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

      5. *-commutative 84.0%

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

      6. div-sub 82.9%

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

      7. sub-neg 82.9%

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

      8. *-commutative 82.9%

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

      9. *-commutative 82.9%

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

      10. times-frac 84.0%

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

   7. Applied egg-rr 84.0%

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

      1. sub-neg 84.0%

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

      2. times-frac 82.9%

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

      3. *-commutative 82.9%

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

      4. div-sub 84.0%

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

      5. associate-/l/ 99.8%

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

      6. div-sub 99.8%

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

      7. *-inverses 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in B around 0 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00021 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1150000 \lor \neg \left(x \leq 4.1 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1150000.0) (not (<= x 4.1e+14)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1150000.0) || !(x <= 4.1e+14)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 - x) / 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 <= (-1150000.0d0)) .or. (.not. (x <= 4.1d+14))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1150000.0) || !(x <= 4.1e+14)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1150000.0) or not (x <= 4.1e+14):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1150000.0) || !(x <= 4.1e+14))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1150000.0) || ~((x <= 4.1e+14)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1150000.0], N[Not[LessEqual[x, 4.1e+14]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e6 or 4.1e14 < 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. cancel-sign-sub-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Step-by-step derivation

      1. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-rgt-neg-out 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-/l* 98.8%

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

   8. Simplified 98.8%

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

      1. distribute-frac-neg 98.8%

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

      2. neg-sub0 98.8%

         \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]

      3. add-sqr-sqrt 78.8%

         \[\leadsto 0 - \frac{\color{blue}{\sqrt{\cos B} \cdot \sqrt{\cos B}}}{\frac{\sin B}{x}} \]

      4. sqrt-unprod 79.0%

         \[\leadsto 0 - \frac{\color{blue}{\sqrt{\cos B \cdot \cos B}}}{\frac{\sin B}{x}} \]

      5. sqr-neg 79.0%

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

      6. sqrt-unprod 0.2%

         \[\leadsto 0 - \frac{\color{blue}{\sqrt{-\cos B} \cdot \sqrt{-\cos B}}}{\frac{\sin B}{x}} \]

      7. add-sqr-sqrt 0.4%

         \[\leadsto 0 - \frac{\color{blue}{-\cos B}}{\frac{\sin B}{x}} \]

      8. associate-/r/ 0.4%

         \[\leadsto 0 - \color{blue}{\frac{-\cos B}{\sin B} \cdot x} \]

      9. clear-num 0.4%

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

      10. add-sqr-sqrt 0.2%

          \[\leadsto 0 - \frac{1}{\frac{\sin B}{\color{blue}{\sqrt{-\cos B} \cdot \sqrt{-\cos B}}}} \cdot x \]

      11. sqrt-unprod 79.0%

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

      12. sqr-neg 79.0%

          \[\leadsto 0 - \frac{1}{\frac{\sin B}{\sqrt{\color{blue}{\cos B \cdot \cos B}}}} \cdot x \]

      13. sqrt-unprod 78.8%

          \[\leadsto 0 - \frac{1}{\frac{\sin B}{\color{blue}{\sqrt{\cos B} \cdot \sqrt{\cos B}}}} \cdot x \]

      14. add-sqr-sqrt 98.7%

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

      15. tan-quot 98.8%

          \[\leadsto 0 - \frac{1}{\color{blue}{\tan B}} \cdot x \]

   10. Applied egg-rr 98.8%

       \[\leadsto \color{blue}{0 - \frac{1}{\tan B} \cdot x} \]
   11. Step-by-step derivation

       1. neg-sub0 98.8%

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

       2. associate-*l/ 99.0%

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

       3. *-lft-identity 99.0%

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

       4. distribute-neg-frac 99.0%

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

   12. Simplified 99.0%

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

   if -1.15e6 < x < 4.1e14

   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. cancel-sign-sub-inv 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Step-by-step derivation

      1. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. inv-pow 99.8%

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

      2. frac-sub 82.9%

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

      3. *-un-lft-identity 82.9%

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

      4. *-commutative 82.9%

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

      5. *-commutative 82.9%

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

      6. div-sub 81.8%

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

      7. sub-neg 81.8%

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

      8. *-commutative 81.8%

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

      9. *-commutative 81.8%

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

      10. times-frac 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. sub-neg 82.9%

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

      2. times-frac 81.8%

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

      3. *-commutative 81.8%

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

      4. div-sub 82.9%

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

      5. associate-/l/ 99.8%

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

      6. div-sub 99.8%

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

      7. *-inverses 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in B around 0 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1150000 \lor \neg \left(x \leq 4.1 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 6: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.3) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.3) || !(x <= 1.0)) {
		tmp = -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 <= (-1.3d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = -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 <= -1.3) || !(x <= 1.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.3) or not (x <= 1.0):
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.3) || !(x <= 1.0))
		tmp = Float64(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 <= -1.3) || ~((x <= 1.0)))
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.3], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004 or 1 < 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. cancel-sign-sub-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Step-by-step derivation

      1. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-rgt-neg-out 98.0%

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

      4. *-commutative 98.0%

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

      5. associate-/l* 97.9%

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

   8. Simplified 97.9%

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

      1. distribute-frac-neg 97.9%

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

      2. neg-sub0 97.9%

         \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]

      3. add-sqr-sqrt 78.3%

         \[\leadsto 0 - \frac{\color{blue}{\sqrt{\cos B} \cdot \sqrt{\cos B}}}{\frac{\sin B}{x}} \]

      4. sqrt-unprod 78.6%

         \[\leadsto 0 - \frac{\color{blue}{\sqrt{\cos B \cdot \cos B}}}{\frac{\sin B}{x}} \]

      5. sqr-neg 78.6%

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

      6. sqrt-unprod 0.2%

         \[\leadsto 0 - \frac{\color{blue}{\sqrt{-\cos B} \cdot \sqrt{-\cos B}}}{\frac{\sin B}{x}} \]

      7. add-sqr-sqrt 0.4%

         \[\leadsto 0 - \frac{\color{blue}{-\cos B}}{\frac{\sin B}{x}} \]

      8. associate-/r/ 0.4%

         \[\leadsto 0 - \color{blue}{\frac{-\cos B}{\sin B} \cdot x} \]

      9. clear-num 0.4%

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

      10. add-sqr-sqrt 0.2%

          \[\leadsto 0 - \frac{1}{\frac{\sin B}{\color{blue}{\sqrt{-\cos B} \cdot \sqrt{-\cos B}}}} \cdot x \]

      11. sqrt-unprod 78.5%

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

      12. sqr-neg 78.5%

          \[\leadsto 0 - \frac{1}{\frac{\sin B}{\sqrt{\color{blue}{\cos B \cdot \cos B}}}} \cdot x \]

      13. sqrt-unprod 78.4%

          \[\leadsto 0 - \frac{1}{\frac{\sin B}{\color{blue}{\sqrt{\cos B} \cdot \sqrt{\cos B}}}} \cdot x \]

      14. add-sqr-sqrt 97.9%

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

      15. tan-quot 98.0%

          \[\leadsto 0 - \frac{1}{\color{blue}{\tan B}} \cdot x \]

   10. Applied egg-rr 98.0%

       \[\leadsto \color{blue}{0 - \frac{1}{\tan B} \cdot x} \]
   11. Step-by-step derivation

       1. neg-sub0 98.0%

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

       2. associate-*l/ 98.1%

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

       3. *-lft-identity 98.1%

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

       4. distribute-neg-frac 98.1%

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

   12. Simplified 98.1%

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

   if -1.30000000000000004 < x < 1

   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. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 73.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -220000000000 \lor \neg \left(B \leq 1.5 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \left(0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -220000000000.0) (not (<= B 1.5e+32)))
   (/ 1.0 (sin B))
   (+ (/ 1.0 B) (- (* 0.3333333333333333 (* B x)) (/ x B)))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -220000000000.0) || !(B <= 1.5e+32)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 / B) + ((0.3333333333333333 * (B * x)) - (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 ((b <= (-220000000000.0d0)) .or. (.not. (b <= 1.5d+32))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 / b) + ((0.3333333333333333d0 * (b * x)) - (x / b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -220000000000.0) || !(B <= 1.5e+32)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 / B) + ((0.3333333333333333 * (B * x)) - (x / B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -220000000000.0) or not (B <= 1.5e+32):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 / B) + ((0.3333333333333333 * (B * x)) - (x / B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -220000000000.0) || !(B <= 1.5e+32))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) + Float64(Float64(0.3333333333333333 * Float64(B * x)) - Float64(x / B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -220000000000.0) || ~((B <= 1.5e+32)))
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 / B) + ((0.3333333333333333 * (B * x)) - (x / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -220000000000.0], N[Not[LessEqual[B, 1.5e+32]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -2.2e11 or 1.5e32 < B

   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. distribute-lft-neg-in 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 53.1%

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

   if -2.2e11 < B < 1.5e32

   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. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 99.8%

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

   5. Taylor expanded in B around 0 94.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -220000000000 \lor \neg \left(B \leq 1.5 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \left(0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{x}{B}\right)\\ \end{array} \]

Alternative 8: 51.0% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(1.0 / B), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] - N[(x / 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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 74.4%

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

5. Taylor expanded in B around 0 49.8%

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

6. Final simplification 49.8%

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

Alternative 9: 51.0% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(x * N[(B * 0.3333333333333333), $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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 74.4%

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

5. Taylor expanded in B around 0 49.8%

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

   1. neg-mul-1 49.8%

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

   2. +-commutative 49.8%

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

   3. associate-+r+ 49.8%

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

   4. +-commutative 49.8%

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

   5. sub-neg 49.8%

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

   6. div-sub 49.8%

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

   7. associate-*r* 49.8%

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

7. Simplified 49.8%

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

8. Final simplification 49.8%

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

Alternative 10: 49.6% accurate, 25.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. 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. distribute-lft-neg-in 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 53.6%

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

      1. neg-mul-1 53.6%

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

      2. sub-neg 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in x around inf 52.5%

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

      1. neg-mul-1 52.5%

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

      2. distribute-neg-frac 52.5%

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

   9. Simplified 52.5%

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

   if -1 < x < 1

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. 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. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 44.5%

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

   5. Taylor expanded in B around 0 44.6%

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

   6. Taylor expanded in x around 0 43.5%

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

4. Final simplification 48.4%

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

Alternative 11: 50.7% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(1.0 / 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. 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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 49.4%

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

   1. neg-mul-1 49.4%

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

   2. sub-neg 49.4%

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

6. Simplified 49.4%

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

   1. div-sub 49.4%

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

8. Applied egg-rr 49.4%

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

9. Final simplification 49.4%

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

Alternative 12: 50.7% 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. 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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 49.4%

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

   1. neg-mul-1 49.4%

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

   2. sub-neg 49.4%

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

6. Simplified 49.4%

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

7. Final simplification 49.4%

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

Alternative 13: 26.2% 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. 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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 74.4%

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

5. Taylor expanded in B around 0 49.8%

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

6. Taylor expanded in x around 0 21.3%

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

7. Final simplification 21.3%

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

Reproduce

herbie shell --seed 2023313 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))