VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. 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 2: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -195000000 \lor \neg \left(x \leq 2600000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -195000000.0) (not (<= x 2600000.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -195000000.0) || !(x <= 2600000.0)) {
		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 <= (-195000000.0d0)) .or. (.not. (x <= 2600000.0d0))) 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 <= -195000000.0) || !(x <= 2600000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -195000000.0) or not (x <= 2600000.0):
		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 <= -195000000.0) || !(x <= 2600000.0))
		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 <= -195000000.0) || ~((x <= 2600000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -195000000.0], N[Not[LessEqual[x, 2600000.0]], $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 < -1.95e8 or 2.6e6 < 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. tan-quot 99.6%

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

      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%

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

      1. div-sub 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. mul-1-neg 98.9%

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

       2. associate-/l* 98.9%

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

       3. distribute-neg-frac 98.9%

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

   11. Simplified 98.9%

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

       1. tan-quot 99.0%

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

       2. expm1-log1p-u 87.2%

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

       3. expm1-udef 52.6%

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

   13. Applied egg-rr 52.6%

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

       1. expm1-def 87.2%

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

       2. expm1-log1p 99.0%

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

   15. Simplified 99.0%

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

   if -1.95e8 < x < 2.6e6

   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. 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%

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

      1. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -195000000 \lor \neg \left(x \leq 2600000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 3: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.9) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.9) || !(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 <= (-1.9d0)) .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 <= -1.9) || !(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 <= -1.9) 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 <= -1.9) || !(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 <= -1.9) || ~((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, -1.9], 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 < -1.8999999999999999 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. tan-quot 99.6%

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

      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%

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

      1. div-sub 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 97.4%

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

       1. mul-1-neg 97.4%

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

       2. associate-/l* 97.4%

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

       3. distribute-neg-frac 97.4%

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

   11. Simplified 97.4%

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

       1. tan-quot 97.5%

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

       2. expm1-log1p-u 86.0%

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

       3. expm1-udef 51.5%

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

   13. Applied egg-rr 51.5%

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

       1. expm1-def 86.0%

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

       2. expm1-log1p 97.5%

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

   15. Simplified 97.5%

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

   if -1.8999999999999999 < 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 98.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 74.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 470000000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot -0.16666666666666666 - x \cdot -0.5\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -5e-7)
   (+ (/ (- 1.0 x) B) (* B -0.16666666666666666))
   (if (<= x 470000000.0)
     (/ 1.0 (sin B))
     (- (* B (- (* x -0.16666666666666666) (* x -0.5))) (/ x B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -5e-7) {
		tmp = ((1.0 - x) / B) + (B * -0.16666666666666666);
	} else if (x <= 470000000.0) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * ((x * -0.16666666666666666) - (x * -0.5))) - (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 <= (-5d-7)) then
        tmp = ((1.0d0 - x) / b) + (b * (-0.16666666666666666d0))
    else if (x <= 470000000.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * ((x * (-0.16666666666666666d0)) - (x * (-0.5d0)))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -5e-7) {
		tmp = ((1.0 - x) / B) + (B * -0.16666666666666666);
	} else if (x <= 470000000.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * ((x * -0.16666666666666666) - (x * -0.5))) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -5e-7:
		tmp = ((1.0 - x) / B) + (B * -0.16666666666666666)
	elif x <= 470000000.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * ((x * -0.16666666666666666) - (x * -0.5))) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -5e-7)
		tmp = Float64(Float64(Float64(1.0 - x) / B) + Float64(B * -0.16666666666666666));
	elseif (x <= 470000000.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * Float64(Float64(x * -0.16666666666666666) - Float64(x * -0.5))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -5e-7)
		tmp = ((1.0 - x) / B) + (B * -0.16666666666666666);
	elseif (x <= 470000000.0)
		tmp = 1.0 / sin(B);
	else
		tmp = (B * ((x * -0.16666666666666666) - (x * -0.5))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -5e-7], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 470000000.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(N[(x * -0.16666666666666666), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999977e-7

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. distribute-lft-neg-in 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in B around 0 54.3%

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

   5. Taylor expanded in B around 0 53.7%

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

      1. neg-mul-1 53.7%

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

      2. +-commutative 53.7%

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

      3. associate-+r+ 53.7%

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

      4. +-commutative 53.7%

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

      5. sub-neg 53.7%

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

      6. div-sub 53.8%

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

      7. *-commutative 53.8%

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

   7. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B} + B \cdot 0.16666666666666666} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 23.6%

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

      2. sqrt-unprod 53.4%

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

      3. *-commutative 53.4%

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

      4. *-commutative 53.4%

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

      5. swap-sqr 53.4%

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

      6. metadata-eval 53.4%

         \[\leadsto \frac{1 - x}{B} + \sqrt{\color{blue}{0.027777777777777776} \cdot \left(B \cdot B\right)} \]

   9. Applied egg-rr 53.4%

      \[\leadsto \frac{1 - x}{B} + \color{blue}{\sqrt{0.027777777777777776 \cdot \left(B \cdot B\right)}} \]
   10. Step-by-step derivation

       1. unpow2 53.4%

          \[\leadsto \frac{1 - x}{B} + \sqrt{0.027777777777777776 \cdot \color{blue}{{B}^{2}}} \]

       2. *-commutative 53.4%

          \[\leadsto \frac{1 - x}{B} + \sqrt{\color{blue}{{B}^{2} \cdot 0.027777777777777776}} \]

       3. unpow2 53.4%

          \[\leadsto \frac{1 - x}{B} + \sqrt{\color{blue}{\left(B \cdot B\right)} \cdot 0.027777777777777776} \]

   11. Simplified 53.4%

       \[\leadsto \frac{1 - x}{B} + \color{blue}{\sqrt{\left(B \cdot B\right) \cdot 0.027777777777777776}} \]

   12. Taylor expanded in B around -inf 54.0%

       \[\leadsto \frac{1 - x}{B} + \color{blue}{-0.16666666666666666 \cdot B} \]
   13. Step-by-step derivation

       1. *-commutative 54.0%

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

   14. Simplified 54.0%

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

   if -4.99999999999999977e-7 < x < 4.7e8

   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 99.1%

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

   if 4.7e8 < 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/ 99.7%

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

      7. *-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. Step-by-step derivation

      1. tan-quot 99.6%

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

      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 99.6%

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

      1. div-sub 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around inf 99.6%

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

       1. mul-1-neg 99.6%

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

       2. associate-/l* 99.6%

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

       3. distribute-neg-frac 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in B around 0 55.1%

       \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)\right) + -1 \cdot \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 470000000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot -0.16666666666666666 - x \cdot -0.5\right) - \frac{x}{B}\\ \end{array} \]

Alternative 5: 51.8% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((1.0d0 - x) / b) + (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))
end function

Java

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

Python

def code(B, x):
	return ((1.0 - x) / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))

Julia

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

MATLAB

function tmp = code(B, x)
	tmp = ((1.0 - x) / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. 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 52.9%

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

   1. +-commutative 52.9%

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

   2. mul-1-neg 52.9%

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

   3. sub-neg 52.9%

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

   4. associate--l+ 52.9%

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

   5. *-commutative 52.9%

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

   6. div-sub 52.9%

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

6. Simplified 52.9%

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

7. Final simplification 52.9%

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

Alternative 6: 51.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{B} + B \cdot -0.16666666666666666 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((1.0d0 - x) / b) + (b * (-0.16666666666666666d0))
end function

Java

public static double code(double B, double x) {
	return ((1.0 - x) / B) + (B * -0.16666666666666666);
}

Python

def code(B, x):
	return ((1.0 - x) / B) + (B * -0.16666666666666666)

Julia

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

MATLAB

function tmp = code(B, x)
	tmp = ((1.0 - x) / B) + (B * -0.16666666666666666);
end

Wolfram

code[B_, x_] := N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * -0.16666666666666666), $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 76.4%

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

5. Taylor expanded in B around 0 52.7%

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

   1. neg-mul-1 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r+ 52.7%

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

   4. +-commutative 52.7%

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

   5. sub-neg 52.7%

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

   6. div-sub 52.7%

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

   7. *-commutative 52.7%

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

7. Simplified 52.7%

   \[\leadsto \color{blue}{\frac{1 - x}{B} + B \cdot 0.16666666666666666} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 27.8%

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

   2. sqrt-unprod 52.4%

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

   3. *-commutative 52.4%

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

   4. *-commutative 52.4%

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

   5. swap-sqr 52.4%

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

   6. metadata-eval 52.4%

      \[\leadsto \frac{1 - x}{B} + \sqrt{\color{blue}{0.027777777777777776} \cdot \left(B \cdot B\right)} \]

9. Applied egg-rr 52.4%

   \[\leadsto \frac{1 - x}{B} + \color{blue}{\sqrt{0.027777777777777776 \cdot \left(B \cdot B\right)}} \]
10. Step-by-step derivation

    1. unpow2 52.4%

       \[\leadsto \frac{1 - x}{B} + \sqrt{0.027777777777777776 \cdot \color{blue}{{B}^{2}}} \]

    2. *-commutative 52.4%

       \[\leadsto \frac{1 - x}{B} + \sqrt{\color{blue}{{B}^{2} \cdot 0.027777777777777776}} \]

    3. unpow2 52.4%

       \[\leadsto \frac{1 - x}{B} + \sqrt{\color{blue}{\left(B \cdot B\right)} \cdot 0.027777777777777776} \]

11. Simplified 52.4%

    \[\leadsto \frac{1 - x}{B} + \color{blue}{\sqrt{\left(B \cdot B\right) \cdot 0.027777777777777776}} \]

12. Taylor expanded in B around -inf 52.7%

    \[\leadsto \frac{1 - x}{B} + \color{blue}{-0.16666666666666666 \cdot B} \]
13. Step-by-step derivation

    1. *-commutative 52.7%

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

14. Simplified 52.7%

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

15. Final simplification 52.7%

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

Alternative 7: 49.7% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.28 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.28e-27))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.28e-27)) {
		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 <= 1.28d-27))) 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 <= 1.28e-27)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.28e-27):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.28e-27))
		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 <= 1.28e-27)))
		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, 1.28e-27]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.27999999999999993e-27 < 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 52.3%

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

      1. neg-mul-1 52.3%

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

      2. sub-neg 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in x around inf 51.3%

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

      1. neg-mul-1 51.5%

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

   9. Simplified 51.3%

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

   if -1 < x < 1.27999999999999993e-27

   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 53.1%

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

      1. neg-mul-1 53.1%

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

      2. sub-neg 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in x around 0 52.1%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.28 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 8: 51.6% 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 52.7%

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

   1. neg-mul-1 52.7%

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

   2. sub-neg 52.7%

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

6. Simplified 52.7%

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

7. Final simplification 52.7%

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

Alternative 9: 3.1% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. 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 76.4%

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

5. Taylor expanded in B around 0 52.7%

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

   1. neg-mul-1 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r+ 52.7%

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

   4. +-commutative 52.7%

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

   5. sub-neg 52.7%

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

   6. div-sub 52.7%

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

   7. *-commutative 52.7%

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

7. Simplified 52.7%

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

8. Taylor expanded in x around inf 27.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} + B \cdot 0.16666666666666666 \]
9. Step-by-step derivation

   1. neg-mul-1 27.7%

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

10. Simplified 27.7%

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

11. Taylor expanded in x around 0 3.0%

    \[\leadsto \color{blue}{0.16666666666666666 \cdot B} \]
12. Step-by-step derivation

    1. *-commutative 3.0%

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

13. Simplified 3.0%

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

14. Final simplification 3.0%

    \[\leadsto B \cdot 0.16666666666666666 \]

Alternative 10: 27.0% 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 52.7%

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

   1. neg-mul-1 52.7%

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

   2. sub-neg 52.7%

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

6. Simplified 52.7%

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

7. Taylor expanded in x around 0 27.4%

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

8. Final simplification 27.4%

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

Reproduce

herbie shell --seed 2023272 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))