VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.8%

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

   10. *-rgt-identity 99.8%

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

   11. tan-neg 99.8%

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

   12. distribute-neg-frac2 99.8%

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

   13. distribute-neg-frac 99.8%

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

   14. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{-16}:\\ \;\;\;\;t\_0 \cdot \left(\frac{1}{x} + -1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\sin B - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= x -3.3e-16)
     (* t_0 (+ (/ 1.0 x) -1.0))
     (if (<= x 1.0) (/ 1.0 (sin B)) (- (sin B) t_0)))))

C

double code(double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (x <= -3.3e-16) {
		tmp = t_0 * ((1.0 / x) + -1.0);
	} else if (x <= 1.0) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = sin(B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (x <= (-3.3d-16)) then
        tmp = t_0 * ((1.0d0 / x) + (-1.0d0))
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = sin(b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (x <= -3.3e-16) {
		tmp = t_0 * ((1.0 / x) + -1.0);
	} else if (x <= 1.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = Math.sin(B) - t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if x <= -3.3e-16:
		tmp = t_0 * ((1.0 / x) + -1.0)
	elif x <= 1.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = math.sin(B) - t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (x <= -3.3e-16)
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) + -1.0));
	elseif (x <= 1.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(sin(B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (x <= -3.3e-16)
		tmp = t_0 * ((1.0 / x) + -1.0);
	elseif (x <= 1.0)
		tmp = 1.0 / sin(B);
	else
		tmp = sin(B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.3e-16], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[Sin[B], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.29999999999999988e-16

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-frac-neg2 99.6%

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

      6. tan-neg 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. 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/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/r/ 99.8%

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

      2. tan-quot 99.8%

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

      3. clear-num 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. frac-sub 82.0%

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

      2. div-inv 79.6%

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

      3. *-un-lft-identity 79.6%

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

      4. *-rgt-identity 79.6%

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

      5. metadata-eval 79.6%

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

      6. frac-times 79.5%

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

      7. associate-*l/ 79.7%

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

      8. *-un-lft-identity 79.7%

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

      9. clear-num 79.8%

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

   10. Applied egg-rr 79.8%

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

       1. associate-*r/ 99.8%

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

       2. associate-*l/ 99.8%

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

       3. div-sub 99.9%

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

       4. sub-neg 99.9%

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

       5. associate-/l/ 99.8%

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

       6. *-commutative 99.8%

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

       7. *-inverses 99.8%

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

       8. metadata-eval 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in B around 0 99.9%

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

   if -3.29999999999999988e-16 < x < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 98.8%

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

   if 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. *-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-frac-neg2 99.6%

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

      6. tan-neg 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

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

   6. Applied egg-rr 99.7%

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

      1. associate-/r/ 99.6%

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

      2. tan-quot 99.8%

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

      3. sub-neg 99.8%

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

      4. add-exp-log 45.5%

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

      5. neg-log 45.5%

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

      6. add-sqr-sqrt 45.5%

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

      7. sqrt-unprod 45.5%

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

      8. sqr-neg 45.5%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 42.6%

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

      11. add-exp-log 95.4%

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

      12. distribute-neg-frac2 95.4%

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

   8. Applied egg-rr 95.4%

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

      1. distribute-frac-neg2 95.4%

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

      2. sub-neg 95.4%

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

   10. Simplified 95.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{\tan B} \cdot \left(\frac{1}{x} + -1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\sin B - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-16} \lor \neg \left(x \leq 1.02\right):\\ \;\;\;\;\frac{x}{\tan B} \cdot \left(\frac{1}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.3e-16) (not (<= x 1.02)))
   (* (/ x (tan B)) (+ (/ 1.0 x) -1.0))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.3e-16) || !(x <= 1.02)) {
		tmp = (x / tan(B)) * ((1.0 / x) + -1.0);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.3d-16)) .or. (.not. (x <= 1.02d0))) then
        tmp = (x / tan(b)) * ((1.0d0 / x) + (-1.0d0))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -3.3e-16) || !(x <= 1.02)) {
		tmp = (x / Math.tan(B)) * ((1.0 / x) + -1.0);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.3e-16) or not (x <= 1.02):
		tmp = (x / math.tan(B)) * ((1.0 / x) + -1.0)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3.3e-16) || !(x <= 1.02))
		tmp = Float64(Float64(x / tan(B)) * Float64(Float64(1.0 / x) + -1.0));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -3.3e-16) || ~((x <= 1.02)))
		tmp = (x / tan(B)) * ((1.0 / x) + -1.0);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.3e-16], N[Not[LessEqual[x, 1.02]], $MachinePrecision]], N[(N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.29999999999999988e-16 or 1.02 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. distribute-frac-neg2 99.7%

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

      6. tan-neg 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

      1. tan-quot 99.7%

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

      2. associate-/r/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/r/ 99.7%

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

      2. tan-quot 99.8%

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

      3. clear-num 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. frac-sub 79.4%

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

      2. div-inv 77.5%

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

      3. *-un-lft-identity 77.5%

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

      4. *-rgt-identity 77.5%

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

      5. metadata-eval 77.5%

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

      6. frac-times 77.4%

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

      7. associate-*l/ 77.5%

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

      8. *-un-lft-identity 77.5%

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

      9. clear-num 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. associate-*r/ 99.8%

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

       2. associate-*l/ 99.8%

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

       3. div-sub 99.8%

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

       4. sub-neg 99.8%

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

       5. associate-/l/ 99.8%

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

       6. *-commutative 99.8%

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

       7. *-inverses 99.8%

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

       8. metadata-eval 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in B around 0 98.3%

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

   if -3.29999999999999988e-16 < x < 1.02

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-16} \lor \neg \left(x \leq 1.02\right):\\ \;\;\;\;\frac{x}{\tan B} \cdot \left(\frac{1}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-15} \lor \neg \left(x \leq 0.0022\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.45e-15) (not (<= x 0.0022)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.45e-15) || !(x <= 0.0022)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.45d-15)) .or. (.not. (x <= 0.0022d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.45e-15) || !(x <= 0.0022)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.45e-15) or not (x <= 0.0022):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.45e-15) || !(x <= 0.0022))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.45e-15) || ~((x <= 0.0022)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.45e-15], N[Not[LessEqual[x, 0.0022]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000009e-15 or 0.00220000000000000013 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. distribute-frac-neg2 99.7%

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

      6. tan-neg 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 97.6%

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

   if -1.45000000000000009e-15 < x < 0.00220000000000000013

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-15} \lor \neg \left(x \leq 0.0022\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.3) || !(x <= 1.0)) {
		tmp = 1.0 - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.3d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.3) || !(x <= 1.0)) {
		tmp = 1.0 - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004 or 1 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. distribute-frac-neg2 99.7%

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

      6. tan-neg 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

      1. add-exp-log 38.6%

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

      2. log-rec 38.6%

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

   6. Applied egg-rr 38.6%

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

      1. exp-neg 38.6%

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

      2. add-exp-log 99.8%

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

      3. add-sqr-sqrt 38.6%

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

      4. associate-/r* 38.6%

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

      5. metadata-eval 38.6%

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

      6. sqrt-div 38.6%

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

      7. add-exp-log 38.6%

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

      8. neg-log 38.6%

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

      9. add-sqr-sqrt 38.6%

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

      10. sqrt-unprod 38.6%

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

      11. sqr-neg 38.6%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 36.6%

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

      14. add-exp-log 36.6%

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

   8. Applied egg-rr 36.6%

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

      1. *-inverses 95.8%

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

   10. Simplified 95.8%

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

   if -1.30000000000000004 < x < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 98.5%

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

4. Final simplification 97.3%

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

Alternative 6: 63.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 2.8e-5) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 2.8e-5) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 2.8d-5) then
        tmp = (1.0d0 - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 2.8e-5) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 2.8e-5:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 2.8e-5)
		tmp = Float64(Float64(1.0 - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 2.8e-5)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 2.8e-5], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.79999999999999996e-5

   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 70.2%

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

   if 2.79999999999999996e-5 < B

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 63.6%

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

Alternative 7: 50.5% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.02\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.02))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.02)) {
		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 <= 1.02d0))) 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 <= 1.02)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.02):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.02))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.02)))
		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, 1.02]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.02 < x

   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 58.2%

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

   4. Taylor expanded in x around inf 56.3%

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

      1. neg-mul-1 56.3%

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

   6. Simplified 56.3%

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

   if -1 < x < 1.02

   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 51.6%

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

   4. Taylor expanded in x around 0 51.2%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.02\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.4% 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 54.6%

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

Alternative 9: 26.6% 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 54.6%

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

4. Taylor expanded in x around 0 29.1%

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

Reproduce

herbie shell --seed 2024092 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))