VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.4s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. 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. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2100:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 165000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -2100.0)
   (/ (- x) (tan B))
   (if (<= x 165000.0)
     (- (/ 1.0 (sin B)) (/ x B))
     (/ (* x (- (cos B))) (sin B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -2100.0) {
		tmp = -x / tan(B);
	} else if (x <= 165000.0) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (x * -cos(B)) / 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 <= (-2100.0d0)) then
        tmp = -x / tan(b)
    else if (x <= 165000.0d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (x * -cos(b)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -2100.0) {
		tmp = -x / Math.tan(B);
	} else if (x <= 165000.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -2100.0:
		tmp = -x / math.tan(B)
	elif x <= 165000.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (x * -math.cos(B)) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -2100.0)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (x <= 165000.0)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -2100.0)
		tmp = -x / tan(B);
	elseif (x <= 165000.0)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (x * -cos(B)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -2100.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 165000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2100

   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. cancel-sign-sub-inv 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 75.2%

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

   6. Applied egg-rr 75.2%

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

      1. rem-log-exp 99.7%

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

      2. tan-quot 99.6%

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

      3. add-sqr-sqrt 79.1%

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

      4. sqrt-unprod 79.4%

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

      5. sqr-neg 79.4%

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

      6. sqrt-unprod 0.1%

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

      7. add-sqr-sqrt 0.5%

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

      8. associate-/l* 0.5%

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

      9. div-inv 0.5%

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

      10. add-sqr-sqrt 0.1%

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

      11. sqrt-unprod 79.1%

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

      12. sqr-neg 79.1%

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

      13. sqrt-unprod 79.0%

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

      14. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around inf 97.8%

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

       1. associate-*r/ 97.8%

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

       2. neg-mul-1 97.8%

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

       3. distribute-lft-neg-in 97.8%

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

   11. Simplified 97.8%

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

       1. associate-/l* 97.9%

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

       2. neg-sub0 97.9%

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

       3. div-sub 97.9%

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

       4. quot-tan 97.9%

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

       5. quot-tan 98.0%

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

   13. Applied egg-rr 98.0%

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

       1. div-sub 98.0%

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

       2. neg-sub0 98.0%

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

   15. Simplified 98.0%

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

   if -2100 < x < 165000

   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. Add Preprocessing

   5. Taylor expanded in B around 0 99.2%

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

   if 165000 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. rem-log-exp 99.8%

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

      2. tan-quot 99.7%

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

      3. add-sqr-sqrt 74.7%

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

      4. sqrt-unprod 75.0%

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

      5. sqr-neg 75.0%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 0.5%

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

      8. associate-/l* 0.5%

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

      9. div-inv 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 74.8%

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

      12. sqr-neg 74.8%

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

      13. sqrt-unprod 74.5%

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

      14. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around inf 99.3%

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

       1. associate-*r/ 99.3%

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

       2. neg-mul-1 99.3%

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

       3. distribute-lft-neg-in 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2100:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 165000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -760 \lor \neg \left(x \leq 450000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -760.0) (not (<= x 450000.0)))
   (/ (- x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -760.0) || !(x <= 450000.0)) {
		tmp = -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 <= (-760.0d0)) .or. (.not. (x <= 450000.0d0))) then
        tmp = -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 <= -760.0) || !(x <= 450000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -760.0) or not (x <= 450000.0):
		tmp = -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 <= -760.0) || !(x <= 450000.0))
		tmp = Float64(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 <= -760.0) || ~((x <= 450000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -760.0], N[Not[LessEqual[x, 450000.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $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 < -760 or 4.5e5 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. rem-log-exp 99.8%

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

      2. tan-quot 99.7%

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

      3. add-sqr-sqrt 77.0%

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

      4. sqrt-unprod 77.3%

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

      5. sqr-neg 77.3%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 0.5%

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

      8. associate-/l* 0.5%

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

      9. div-inv 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 77.0%

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

      12. sqr-neg 77.0%

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

      13. sqrt-unprod 76.8%

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

      14. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around inf 98.5%

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

       1. associate-*r/ 98.5%

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

       2. neg-mul-1 98.5%

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

       3. distribute-lft-neg-in 98.5%

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

   11. Simplified 98.5%

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

       1. associate-/l* 98.6%

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

       2. neg-sub0 98.6%

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

       3. div-sub 98.6%

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

       4. quot-tan 98.6%

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

       5. quot-tan 98.6%

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

   13. Applied egg-rr 98.6%

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

       1. div-sub 98.6%

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

       2. neg-sub0 98.6%

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

   15. Simplified 98.6%

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

   if -760 < x < 4.5e5

   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. Add Preprocessing

   5. Taylor expanded in B around 0 99.2%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -760 \lor \neg \left(x \leq 450000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2500 \lor \neg \left(x \leq 9000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2500.0) (not (<= x 9000000.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2500.0) || !(x <= 9000000.0)) {
		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 <= (-2500.0d0)) .or. (.not. (x <= 9000000.0d0))) 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 <= -2500.0) || !(x <= 9000000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2500.0) or not (x <= 9000000.0):
		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 <= -2500.0) || !(x <= 9000000.0))
		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 <= -2500.0) || ~((x <= 9000000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2500.0], N[Not[LessEqual[x, 9000000.0]], $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 < -2500 or 9e6 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. rem-log-exp 99.8%

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

      2. tan-quot 99.7%

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

      3. add-sqr-sqrt 77.0%

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

      4. sqrt-unprod 77.3%

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

      5. sqr-neg 77.3%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 0.5%

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

      8. associate-/l* 0.5%

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

      9. div-inv 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 77.0%

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

      12. sqr-neg 77.0%

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

      13. sqrt-unprod 76.8%

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

      14. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around inf 98.5%

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

       1. associate-*r/ 98.5%

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

       2. neg-mul-1 98.5%

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

       3. distribute-lft-neg-in 98.5%

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

   11. Simplified 98.5%

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

       1. associate-/l* 98.6%

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

       2. neg-sub0 98.6%

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

       3. div-sub 98.6%

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

       4. quot-tan 98.6%

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

       5. quot-tan 98.6%

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

   13. Applied egg-rr 98.6%

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

       1. div-sub 98.6%

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

       2. neg-sub0 98.6%

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

   15. Simplified 98.6%

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

   if -2500 < x < 9e6

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. rem-log-exp 99.8%

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

      2. tan-quot 99.8%

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

      3. add-sqr-sqrt 73.3%

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

      4. sqrt-unprod 99.4%

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

      5. sqr-neg 99.4%

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

      6. sqrt-unprod 26.1%

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

      7. add-sqr-sqrt 97.6%

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

      8. associate-/l* 97.6%

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

      9. div-inv 97.6%

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

      10. add-sqr-sqrt 26.1%

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

      11. sqrt-unprod 99.4%

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

      12. sqr-neg 99.4%

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

      13. sqrt-unprod 73.3%

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

      14. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in B around inf 99.8%

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

       1. div-sub 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in B around 0 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2500 \lor \neg \left(x \leq 9000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.9) (not (<= x 1.1))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.9) || !(x <= 1.1)) {
		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 <= 1.1d0))) 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 <= 1.1)) {
		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 <= 1.1):
		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 <= 1.1))
		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 <= 1.1)))
		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, 1.1]], $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 1.1000000000000001 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. rem-log-exp 99.8%

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

      2. tan-quot 99.7%

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

      3. add-sqr-sqrt 77.0%

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

      4. sqrt-unprod 77.3%

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

      5. sqr-neg 77.3%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 0.5%

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

      8. associate-/l* 0.5%

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

      9. div-inv 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 77.0%

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

      12. sqr-neg 77.0%

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

      13. sqrt-unprod 76.8%

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

      14. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around inf 98.5%

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

       1. associate-*r/ 98.5%

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

       2. neg-mul-1 98.5%

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

       3. distribute-lft-neg-in 98.5%

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

   11. Simplified 98.5%

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

       1. associate-/l* 98.6%

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

       2. neg-sub0 98.6%

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

       3. div-sub 98.6%

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

       4. quot-tan 98.6%

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

       5. quot-tan 98.6%

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

   13. Applied egg-rr 98.6%

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

       1. div-sub 98.6%

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

       2. neg-sub0 98.6%

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

   15. Simplified 98.6%

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

   if -1.8999999999999999 < x < 1.1000000000000001

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.1:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 2.1)
   (-
    (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ 1.0 B))
    (/ x B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 2.1) {
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (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 (b <= 2.1d0) then
        tmp = ((b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + (1.0d0 / b)) - (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 (B <= 2.1) {
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 2.1:
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 2.1)
		tmp = Float64(Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(1.0 / B)) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 2.1)
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 2.1], N[(N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.10000000000000009

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 62.1%

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

   if 2.10000000000000009 < B

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 43.1%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.1:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.5% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 45.3%

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

   4. Taylor expanded in x around inf 45.3%

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

      1. mul-1-neg 45.3%

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

      2. distribute-frac-neg 45.3%

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

   6. Simplified 45.3%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 45.6%

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

   4. Taylor expanded in x around 0 44.1%

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

4. Final simplification 44.7%

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

Alternative 8: 51.8% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

5. Taylor expanded in B around 0 71.6%

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

6. Taylor expanded in B around 0 45.5%

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

   1. associate--l+ 45.5%

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

   2. *-commutative 45.5%

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

   3. div-sub 45.5%

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

8. Simplified 45.5%

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

9. Final simplification 45.5%

   \[\leadsto B \cdot 0.16666666666666666 + \frac{1 - x}{B} \]
10. Add Preprocessing

Alternative 9: 51.6% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 45.5%

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

   1. div-sub 45.5%

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

5. Applied egg-rr 45.5%

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

6. Final simplification 45.5%

   \[\leadsto \frac{1}{B} - \frac{x}{B} \]
7. Add Preprocessing

Alternative 10: 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. Add Preprocessing

3. Taylor expanded in B around 0 45.5%

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

4. Final simplification 45.5%

   \[\leadsto \frac{1 - x}{B} \]
5. Add Preprocessing

Alternative 11: 27.2% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 45.5%

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

4. Taylor expanded in x around 0 22.8%

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

5. Final simplification 22.8%

   \[\leadsto \frac{1}{B} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))