VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 10.3s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. B = 6.7595731061954585e+69
  2. x = -3.6686349712799426e+227

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. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2650000000 \lor \neg \left(x \leq 740000000\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2650000000.0) (not (<= x 740000000.0)))
   (/ x (- (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2650000000.0) || !(x <= 740000000.0)) {
		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 <= (-2650000000.0d0)) .or. (.not. (x <= 740000000.0d0))) 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 <= -2650000000.0) || !(x <= 740000000.0)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2650000000.0) or not (x <= 740000000.0):
		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 <= -2650000000.0) || !(x <= 740000000.0))
		tmp = Float64(x / Float64(-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 <= -2650000000.0) || ~((x <= 740000000.0)))
		tmp = x / -tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2650000000.0], N[Not[LessEqual[x, 740000000.0]], $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 < -2.65e9 or 7.4e8 < 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 x around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. associate-/l* 99.5%

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

      3. distribute-lft-neg-in 99.5%

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

   5. Simplified 99.5%

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

      1. clear-num 99.4%

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

      2. tan-quot 99.4%

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

      3. un-div-inv 99.6%

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

      4. add-sqr-sqrt 48.8%

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

      5. sqrt-unprod 36.1%

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

      6. sqr-neg 36.1%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 0.5%

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

      9. frac-2neg 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 35.3%

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

      12. sqr-neg 35.3%

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

      13. sqrt-unprod 50.4%

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

      14. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

   if -2.65e9 < x < 7.4e8

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. clear-num 99.7%

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

      5. frac-sub 88.5%

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

      6. *-un-lft-identity 88.5%

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

      7. *-commutative 88.5%

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

      8. *-un-lft-identity 88.5%

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

   4. Applied egg-rr 88.5%

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

      1. associate-/r* 99.4%

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

      2. div-sub 99.5%

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

      3. *-inverses 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around inf 99.5%

      \[\leadsto \color{blue}{\frac{x \cdot \left(\cos B \cdot \left(\frac{1}{x \cdot \cos B} - 1\right)\right)}{\sin B}} \]
   8. Step-by-step derivation

      1. associate-*r* 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in B around 0 98.0%

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

       1. sub-neg 98.0%

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

       2. metadata-eval 98.0%

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

       3. distribute-rgt-in 98.0%

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

       4. neg-mul-1 98.0%

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

       5. unsub-neg 98.0%

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

       6. lft-mult-inverse 98.1%

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

   12. Simplified 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2650000000 \lor \neg \left(x \leq 740000000\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.15) (not (<= x 1.0)))
   (/ (- 1.0 x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.15) || !(x <= 1.0)) {
		tmp = (1.0 - 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 <= (-1.15d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 - 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 <= -1.15) || !(x <= 1.0)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.15) or not (x <= 1.0):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.15) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 - 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 <= -1.15) || ~((x <= 1.0)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.15], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999 or 1 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.7%

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

      5. frac-sub 86.2%

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

      6. *-un-lft-identity 86.2%

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

      7. *-commutative 86.2%

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

      8. *-un-lft-identity 86.2%

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

   4. Applied egg-rr 86.2%

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

      1. associate-/r* 99.7%

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

      2. div-sub 99.7%

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

      3. *-inverses 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in B around 0 98.9%

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

      1. div-sub 85.1%

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

      2. clear-num 85.2%

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

      3. sub-neg 85.2%

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

      4. div-inv 84.5%

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

      5. clear-num 84.5%

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

   9. Applied egg-rr 84.5%

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

       1. neg-mul-1 84.5%

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

       2. distribute-rgt-in 99.1%

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

       3. *-commutative 99.1%

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

       4. associate-*r/ 99.1%

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

       5. *-commutative 99.1%

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

       6. distribute-rgt-in 99.1%

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

       7. neg-mul-1 99.1%

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

       8. unsub-neg 99.1%

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

       9. lft-mult-inverse 99.1%

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

   11. Simplified 99.1%

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

   if -1.1499999999999999 < x < 1

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.8%

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

      5. frac-sub 89.5%

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

      6. *-un-lft-identity 89.5%

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

      7. *-commutative 89.5%

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

      8. *-un-lft-identity 89.5%

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

   4. Applied egg-rr 89.5%

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

      1. associate-/r* 99.4%

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

      2. div-sub 99.5%

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

      3. *-inverses 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around inf 99.5%

      \[\leadsto \color{blue}{\frac{x \cdot \left(\cos B \cdot \left(\frac{1}{x \cdot \cos B} - 1\right)\right)}{\sin B}} \]
   8. Step-by-step derivation

      1. associate-*r* 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in B around 0 98.6%

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

       1. sub-neg 98.6%

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

       2. metadata-eval 98.6%

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

       3. distribute-rgt-in 98.6%

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

       4. neg-mul-1 98.6%

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

       5. unsub-neg 98.6%

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

       6. lft-mult-inverse 98.7%

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

   12. Simplified 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.42) (not (<= x 1.0))) (/ x (- (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.42) || !(x <= 1.0)) {
		tmp = x / -tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.42d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x / -tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.42) || !(x <= 1.0)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.42) or not (x <= 1.0):
		tmp = x / -math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.42) || !(x <= 1.0))
		tmp = Float64(x / Float64(-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 <= -1.42) || ~((x <= 1.0)))
		tmp = x / -tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.42], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4199999999999999 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 x around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. associate-/l* 96.9%

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

      3. distribute-lft-neg-in 96.9%

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

   5. Simplified 96.9%

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

      1. clear-num 96.8%

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

      2. tan-quot 96.9%

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

      3. un-div-inv 97.1%

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

      4. add-sqr-sqrt 47.7%

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

      5. sqrt-unprod 35.4%

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

      6. sqr-neg 35.4%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 0.5%

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

      9. frac-2neg 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 34.4%

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

      12. sqr-neg 34.4%

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

      13. sqrt-unprod 49.0%

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

      14. add-sqr-sqrt 97.1%

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

   7. Applied egg-rr 97.1%

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

   if -1.4199999999999999 < 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 x around 0 98.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.195:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.195)
   (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.195) {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.195d0) then
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 0.195) {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.195:
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.195)
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 0.195)
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.195], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.19500000000000001

   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 72.0%

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

      1. associate--l+ 72.0%

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

      2. *-commutative 72.0%

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

      3. div-sub 72.0%

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

   5. Simplified 72.0%

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

   6. Taylor expanded in x around inf 72.4%

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

   if 0.19500000000000001 < B

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 46.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.195:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.7% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.75 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.75e-10))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.75e-10)) {
		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 <= 1.75d-10))) 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 <= 1.75e-10)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.75e-10):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.75e-10))
		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 <= 1.75e-10)))
		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, 1.75e-10]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.7499999999999999e-10 < 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 51.6%

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

   4. Taylor expanded in x around inf 49.7%

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

      1. neg-mul-1 49.7%

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

      2. distribute-neg-frac 49.7%

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

   6. Simplified 49.7%

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

   if -1 < x < 1.7499999999999999e-10

   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 58.7%

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

   4. Taylor expanded in x around 0 58.4%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.75 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.3% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $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. Add Preprocessing

3. Taylor expanded in B around 0 54.8%

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

   1. associate--l+ 54.8%

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

   2. *-commutative 54.8%

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

   3. div-sub 54.8%

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

5. Simplified 54.8%

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

6. Taylor expanded in x around inf 55.1%

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

7. Final simplification 55.1%

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

Alternative 8: 52.1% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 54.8%

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

   1. associate--l+ 54.8%

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

   2. *-commutative 54.8%

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

   3. div-sub 54.8%

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

5. Simplified 54.8%

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

6. Taylor expanded in x around 0 54.9%

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

7. Final simplification 54.9%

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

Alternative 9: 52.0% 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 54.9%

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

4. Final simplification 54.9%

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

Alternative 10: 3.1% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-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 66.7%

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

6. Taylor expanded in B around inf 3.2%

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

   1. *-commutative 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

   \[\leadsto B \cdot 0.16666666666666666 \]
10. Add Preprocessing

Alternative 11: 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 54.9%

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

4. Taylor expanded in x around 0 28.3%

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

5. Final simplification 28.3%

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

Reproduce

herbie shell --seed 2024062 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))