VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.7s

Alternatives: 13

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. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-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 2: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.8 \lor \neg \left(x \leq 2.65\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -10.8) (not (<= x 2.65)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -10.8) || !(x <= 2.65)) {
		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 <= (-10.8d0)) .or. (.not. (x <= 2.65d0))) 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 <= -10.8) || !(x <= 2.65)) {
		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 <= -10.8) or not (x <= 2.65):
		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 <= -10.8) || !(x <= 2.65))
		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 <= -10.8) || ~((x <= 2.65)))
		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, -10.8], N[Not[LessEqual[x, 2.65]], $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 < -10.800000000000001 or 2.64999999999999991 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. *-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 98.8%

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

   if -10.800000000000001 < x < 2.64999999999999991

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-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. tan-quot 99.8%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in B around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. div-sub 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 98.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.8 \lor \neg \left(x \leq 2.65\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.8 \lor \neg \left(x \leq 3.6\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -10.8) (not (<= x 3.6)))
   (- (/ 1.0 B) (/ x (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -10.8) || !(x <= 3.6)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (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 <= (-10.8d0)) .or. (.not. (x <= 3.6d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -10.8) || !(x <= 3.6)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -10.8) or not (x <= 3.6):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -10.8) || !(x <= 3.6))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -10.8) || ~((x <= 3.6)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -10.8], N[Not[LessEqual[x, 3.6]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.800000000000001 or 3.60000000000000009 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. *-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 99.6%

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

   if -10.800000000000001 < x < 3.60000000000000009

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-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 97.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.8 \lor \neg \left(x \leq 3.6\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 4: 73.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.185 \lor \neg \left(x \leq 3.4 \cdot 10^{-7}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.185) (not (<= x 3.4e-7)))
   (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.185) || !(x <= 3.4e-7)) {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / 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 <= (-0.185d0)) .or. (.not. (x <= 3.4d-7))) then
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / 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 <= -0.185) || !(x <= 3.4e-7)) {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.185) or not (x <= 3.4e-7):
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.185) || !(x <= 3.4e-7))
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.185) || ~((x <= 3.4e-7)))
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.185], N[Not[LessEqual[x, 3.4e-7]], $MachinePrecision]], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.185 or 3.39999999999999974e-7 < x

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 55.3%

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

      1. +-commutative 55.3%

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

      2. mul-1-neg 55.3%

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

      3. sub-neg 55.3%

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

      4. associate--l+ 55.3%

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

      5. *-commutative 55.3%

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

      6. *-commutative 55.3%

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

      7. div-sub 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in x around inf 55.3%

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

   if -0.185 < x < 3.39999999999999974e-7

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185 \lor \neg \left(x \leq 3.4 \cdot 10^{-7}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 75.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-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. tan-quot 99.7%

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

   2. associate-/r/ 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in B around inf 99.7%

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

   1. *-commutative 99.7%

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

   2. div-sub 99.7%

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

   3. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in B around 0 78.5%

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

10. Final simplification 78.5%

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

Alternative 6: 51.9% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -9.5:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B -9.5)
   (- 1.0 (/ x B))
   (-
    (+ (/ 1.0 B) (* B (+ 0.16666666666666666 (* x 0.3333333333333333))))
    (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= -9.5) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (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 <= (-9.5d0)) then
        tmp = 1.0d0 - (x / b)
    else
        tmp = ((1.0d0 / b) + (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= -9.5) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= -9.5:
		tmp = 1.0 - (x / B)
	else:
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= -9.5)
		tmp = Float64(1.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= -9.5)
		tmp = 1.0 - (x / B);
	else
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, -9.5], N[(1.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -9.5

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 56.1%

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

      1. add-exp-log 37.4%

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

      2. log-rec 37.3%

         \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{B} \]

   6. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{B} \]
   7. Step-by-step derivation

      1. exp-neg 37.4%

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

      2. add-exp-log 56.1%

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

      3. add-sqr-sqrt 37.3%

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

      4. associate-/r* 37.2%

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

      5. metadata-eval 37.2%

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

      6. sqrt-div 37.3%

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

      7. add-exp-log 37.3%

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{e^{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      8. exp-neg 37.3%

         \[\leadsto \frac{\sqrt{\color{blue}{e^{-\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      9. add-sqr-sqrt 37.2%

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{-\log \sin B} \cdot \sqrt{-\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      10. sqrt-unprod 37.3%

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

      11. sqr-neg 37.3%

          \[\leadsto \frac{\sqrt{e^{\sqrt{\color{blue}{\log \sin B \cdot \log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      12. sqrt-unprod 0.0%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{\log \sin B} \cdot \sqrt{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      13. add-sqr-sqrt 9.1%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      14. add-exp-log 9.1%

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

   8. Applied egg-rr 9.1%

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

      1. *-inverses 9.9%

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

   10. Simplified 9.9%

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

   if -9.5 < B

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. *-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 71.3%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.5:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \end{array} \]

Alternative 7: 51.9% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.1:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B -3.1)
   (- 1.0 (/ x B))
   (+ (/ (- 1.0 x) B) (* B (+ 0.16666666666666666 (* x 0.3333333333333333))))))

C

double code(double B, double x) {
	double tmp;
	if (B <= -3.1) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = ((1.0 - x) / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)));
	}
	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 - (x / b)
    else
        tmp = ((1.0d0 - x) / b) + (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= -3.1) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = ((1.0 - x) / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= -3.1:
		tmp = 1.0 - (x / B)
	else:
		tmp = ((1.0 - x) / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= -3.1)
		tmp = Float64(1.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 - x) / B) + Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= -3.1)
		tmp = 1.0 - (x / B);
	else
		tmp = ((1.0 - x) / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, -3.1], N[(1.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -3.10000000000000009

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 56.1%

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

      1. add-exp-log 37.4%

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

      2. log-rec 37.3%

         \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{B} \]

   6. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{B} \]
   7. Step-by-step derivation

      1. exp-neg 37.4%

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

      2. add-exp-log 56.1%

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

      3. add-sqr-sqrt 37.3%

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

      4. associate-/r* 37.2%

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

      5. metadata-eval 37.2%

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

      6. sqrt-div 37.3%

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

      7. add-exp-log 37.3%

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{e^{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      8. exp-neg 37.3%

         \[\leadsto \frac{\sqrt{\color{blue}{e^{-\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      9. add-sqr-sqrt 37.2%

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{-\log \sin B} \cdot \sqrt{-\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      10. sqrt-unprod 37.3%

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

      11. sqr-neg 37.3%

          \[\leadsto \frac{\sqrt{e^{\sqrt{\color{blue}{\log \sin B \cdot \log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      12. sqrt-unprod 0.0%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{\log \sin B} \cdot \sqrt{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      13. add-sqr-sqrt 9.1%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      14. add-exp-log 9.1%

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

   8. Applied egg-rr 9.1%

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

      1. *-inverses 9.9%

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

   10. Simplified 9.9%

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

   if -3.10000000000000009 < B

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 71.3%

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

      1. +-commutative 71.3%

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

      2. mul-1-neg 71.3%

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

      3. sub-neg 71.3%

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

      4. associate--l+ 71.3%

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

      5. *-commutative 71.3%

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

      6. *-commutative 71.3%

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

      7. div-sub 71.3%

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

   6. Simplified 71.3%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.1:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 8: 51.8% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.1:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B -3.1)
   (- 1.0 (/ x B))
   (- (+ (/ 1.0 B) (* B 0.16666666666666666)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= -3.1) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (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 <= (-3.1d0)) then
        tmp = 1.0d0 - (x / b)
    else
        tmp = ((1.0d0 / b) + (b * 0.16666666666666666d0)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= -3.1) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= -3.1:
		tmp = 1.0 - (x / B)
	else:
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= -3.1)
		tmp = Float64(1.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * 0.16666666666666666)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= -3.1)
		tmp = 1.0 - (x / B);
	else
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, -3.1], N[(1.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -3.10000000000000009

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 56.1%

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

      1. add-exp-log 37.4%

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

      2. log-rec 37.3%

         \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{B} \]

   6. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{B} \]
   7. Step-by-step derivation

      1. exp-neg 37.4%

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

      2. add-exp-log 56.1%

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

      3. add-sqr-sqrt 37.3%

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

      4. associate-/r* 37.2%

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

      5. metadata-eval 37.2%

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

      6. sqrt-div 37.3%

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

      7. add-exp-log 37.3%

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{e^{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      8. exp-neg 37.3%

         \[\leadsto \frac{\sqrt{\color{blue}{e^{-\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      9. add-sqr-sqrt 37.2%

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{-\log \sin B} \cdot \sqrt{-\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      10. sqrt-unprod 37.3%

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

      11. sqr-neg 37.3%

          \[\leadsto \frac{\sqrt{e^{\sqrt{\color{blue}{\log \sin B \cdot \log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      12. sqrt-unprod 0.0%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{\log \sin B} \cdot \sqrt{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      13. add-sqr-sqrt 9.1%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      14. add-exp-log 9.1%

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

   8. Applied egg-rr 9.1%

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

      1. *-inverses 9.9%

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

   10. Simplified 9.9%

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

   if -3.10000000000000009 < B

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. *-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 82.9%

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

   5. Taylor expanded in B around 0 71.1%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.1:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \]

Alternative 9: 51.8% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.1:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B -3.1)
   (- 1.0 (/ x B))
   (+ (/ (- 1.0 x) B) (* B 0.16666666666666666))))

C

double code(double B, double x) {
	double tmp;
	if (B <= -3.1) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = ((1.0 - x) / B) + (B * 0.16666666666666666);
	}
	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 - (x / b)
    else
        tmp = ((1.0d0 - x) / b) + (b * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= -3.1) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = ((1.0 - x) / B) + (B * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= -3.1:
		tmp = 1.0 - (x / B)
	else:
		tmp = ((1.0 - x) / B) + (B * 0.16666666666666666)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= -3.1)
		tmp = Float64(1.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 - x) / B) + Float64(B * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= -3.1)
		tmp = 1.0 - (x / B);
	else
		tmp = ((1.0 - x) / B) + (B * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, -3.1], N[(1.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -3.10000000000000009

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 56.1%

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

      1. add-exp-log 37.4%

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

      2. log-rec 37.3%

         \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{B} \]

   6. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{B} \]
   7. Step-by-step derivation

      1. exp-neg 37.4%

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

      2. add-exp-log 56.1%

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

      3. add-sqr-sqrt 37.3%

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

      4. associate-/r* 37.2%

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

      5. metadata-eval 37.2%

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

      6. sqrt-div 37.3%

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

      7. add-exp-log 37.3%

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{e^{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      8. exp-neg 37.3%

         \[\leadsto \frac{\sqrt{\color{blue}{e^{-\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      9. add-sqr-sqrt 37.2%

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{-\log \sin B} \cdot \sqrt{-\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      10. sqrt-unprod 37.3%

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

      11. sqr-neg 37.3%

          \[\leadsto \frac{\sqrt{e^{\sqrt{\color{blue}{\log \sin B \cdot \log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      12. sqrt-unprod 0.0%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{\log \sin B} \cdot \sqrt{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      13. add-sqr-sqrt 9.1%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      14. add-exp-log 9.1%

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

   8. Applied egg-rr 9.1%

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

      1. *-inverses 9.9%

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

   10. Simplified 9.9%

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

   if -3.10000000000000009 < B

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. *-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 82.9%

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

   5. Taylor expanded in B around 0 71.1%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   6. Step-by-step derivation

      1. associate--l+ 71.1%

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

      2. *-commutative 71.1%

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

      3. div-sub 71.1%

         \[\leadsto B \cdot 0.16666666666666666 + \color{blue}{\frac{1 - x}{B}} \]

   7. Simplified 71.1%

      \[\leadsto \color{blue}{B \cdot 0.16666666666666666 + \frac{1 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.1:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot 0.16666666666666666\\ \end{array} \]

Alternative 10: 49.7% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-5} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.8e-5) (not (<= x 1.0))) (- 1.0 (/ x B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.8e-5) || !(x <= 1.0)) {
		tmp = 1.0 - (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 <= (-3.8d-5)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 - (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 <= -3.8e-5) || !(x <= 1.0)) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.8e-5) or not (x <= 1.0):
		tmp = 1.0 - (x / B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3.8e-5) || !(x <= 1.0))
		tmp = Float64(1.0 - 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 <= -3.8e-5) || ~((x <= 1.0)))
		tmp = 1.0 - (x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.8e-5], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000002e-5 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. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 53.5%

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

      1. add-exp-log 28.2%

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

      2. log-rec 28.2%

         \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{B} \]

   6. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{B} \]
   7. Step-by-step derivation

      1. exp-neg 28.2%

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

      2. add-exp-log 53.5%

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

      3. add-sqr-sqrt 28.2%

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

      4. associate-/r* 28.2%

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

      5. metadata-eval 28.2%

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

      6. sqrt-div 28.2%

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

      7. add-exp-log 28.2%

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{e^{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      8. exp-neg 28.2%

         \[\leadsto \frac{\sqrt{\color{blue}{e^{-\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      9. add-sqr-sqrt 28.2%

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{-\log \sin B} \cdot \sqrt{-\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      10. sqrt-unprod 28.2%

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

      11. sqr-neg 28.2%

          \[\leadsto \frac{\sqrt{e^{\sqrt{\color{blue}{\log \sin B \cdot \log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      12. sqrt-unprod 0.0%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{\log \sin B} \cdot \sqrt{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      13. add-sqr-sqrt 27.3%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      14. add-exp-log 27.3%

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

   8. Applied egg-rr 27.3%

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

      1. *-inverses 51.1%

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

   10. Simplified 51.1%

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

   if -3.8000000000000002e-5 < 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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 56.4%

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

      1. mul-1-neg 56.4%

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

      2. sub-neg 56.4%

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

   6. Simplified 56.4%

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

   7. Taylor expanded in x around 0 55.3%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-5} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 11: 49.6% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.18 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.18) (not (<= x 1.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.18) || !(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 <= (-0.18d0)) .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 <= -0.18) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.18) 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 <= -0.18) || !(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 <= -0.18) || ~((x <= 1.0)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.18], 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 < -0.17999999999999999 or 1 < x

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 53.3%

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

      1. mul-1-neg 53.3%

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

      2. sub-neg 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in x around inf 51.0%

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

      1. neg-mul-1 51.0%

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

      2. distribute-neg-frac 51.0%

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

   9. Simplified 51.0%

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

   if -0.17999999999999999 < 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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 55.6%

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

      1. mul-1-neg 55.6%

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

      2. sub-neg 55.6%

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

   6. Simplified 55.6%

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

   7. Taylor expanded in x around 0 54.6%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.18 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 12: 51.7% accurate, 29.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.1:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B -3.1) (- 1.0 (/ x B)) (/ (- 1.0 x) B)))

C

double code(double B, double x) {
	double tmp;
	if (B <= -3.1) {
		tmp = 1.0 - (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 (b <= (-3.1d0)) then
        tmp = 1.0d0 - (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 (B <= -3.1) {
		tmp = 1.0 - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if B < -3.10000000000000009

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 56.1%

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

      1. add-exp-log 37.4%

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

      2. log-rec 37.3%

         \[\leadsto e^{\color{blue}{-\log \sin B}} - \frac{x}{B} \]

   6. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{e^{-\log \sin B}} - \frac{x}{B} \]
   7. Step-by-step derivation

      1. exp-neg 37.4%

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

      2. add-exp-log 56.1%

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

      3. add-sqr-sqrt 37.3%

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

      4. associate-/r* 37.2%

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

      5. metadata-eval 37.2%

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

      6. sqrt-div 37.3%

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

      7. add-exp-log 37.3%

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{e^{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      8. exp-neg 37.3%

         \[\leadsto \frac{\sqrt{\color{blue}{e^{-\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      9. add-sqr-sqrt 37.2%

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{-\log \sin B} \cdot \sqrt{-\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      10. sqrt-unprod 37.3%

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

      11. sqr-neg 37.3%

          \[\leadsto \frac{\sqrt{e^{\sqrt{\color{blue}{\log \sin B \cdot \log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      12. sqrt-unprod 0.0%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\sqrt{\log \sin B} \cdot \sqrt{\log \sin B}}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      13. add-sqr-sqrt 9.1%

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log \sin B}}}}{\sqrt{\sin B}} - \frac{x}{B} \]

      14. add-exp-log 9.1%

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

   8. Applied egg-rr 9.1%

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

      1. *-inverses 9.9%

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

   10. Simplified 9.9%

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

   if -3.10000000000000009 < B

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. sub-neg 70.6%

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

   6. Simplified 70.6%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.1:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 13: 25.3% 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%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 54.4%

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

   1. mul-1-neg 54.4%

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

   2. sub-neg 54.4%

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

6. Simplified 54.4%

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

7. Taylor expanded in x around 0 29.1%

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

8. Final simplification 29.1%

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

Reproduce

herbie shell --seed 2023217 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))