VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.7s

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. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200000 \lor \neg \left(x \leq 145\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{B \cdot \left(1 - x\right)}{\sin B}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -200000.0) (not (<= x 145.0)))
   (/ (- x) (tan B))
   (/ (/ (* B (- 1.0 x)) (sin B)) B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -200000.0) || !(x <= 145.0)) {
		tmp = -x / tan(B);
	} else {
		tmp = ((B * (1.0 - x)) / sin(B)) / 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 <= (-200000.0d0)) .or. (.not. (x <= 145.0d0))) then
        tmp = -x / tan(b)
    else
        tmp = ((b * (1.0d0 - x)) / sin(b)) / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -200000.0) || !(x <= 145.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = ((B * (1.0 - x)) / Math.sin(B)) / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -200000.0) or not (x <= 145.0):
		tmp = -x / math.tan(B)
	else:
		tmp = ((B * (1.0 - x)) / math.sin(B)) / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -200000.0) || !(x <= 145.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(Float64(B * Float64(1.0 - x)) / sin(B)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -200000.0) || ~((x <= 145.0)))
		tmp = -x / tan(B);
	else
		tmp = ((B * (1.0 - x)) / sin(B)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -200000.0], N[Not[LessEqual[x, 145.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(B * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e5 or 145 < 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%

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

   3. Simplified 99.6%

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

      1. un-div-inv 99.7%

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

      2. add-sqr-sqrt 49.8%

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

      3. sqrt-unprod 41.3%

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

      4. sqr-neg 41.3%

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

      5. sqrt-unprod 0.2%

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

      6. add-sqr-sqrt 0.5%

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

      7. frac-2neg 0.5%

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

      8. div-inv 0.5%

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

      9. fma-def 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 31.9%

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

      12. sqr-neg 31.9%

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

      13. sqrt-unprod 49.6%

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

      14. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 97.5%

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

      1. mul-1-neg 97.5%

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

      2. associate-/l* 97.4%

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

   8. Simplified 97.4%

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

      1. associate-/r/ 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. *-commutative 97.5%

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

      4. *-commutative 97.5%

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

      5. clear-num 97.5%

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

      6. un-div-inv 97.5%

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

      7. quot-tan 97.6%

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

   10. Applied egg-rr 97.6%

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

   if -2e5 < x < 145

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 98.9%

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

      1. frac-sub 65.7%

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

      2. associate-/r* 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. *-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in B around 0 99.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000 \lor \neg \left(x \leq 145\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{B \cdot \left(1 - x\right)}{\sin B}}{B}\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -125000 \lor \neg \left(x \leq 145\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -125000.0) (not (<= x 145.0)))
   (/ (- x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -125000.0) || !(x <= 145.0)) {
		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 <= (-125000.0d0)) .or. (.not. (x <= 145.0d0))) 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 <= -125000.0) || !(x <= 145.0)) {
		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 <= -125000.0) or not (x <= 145.0):
		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 <= -125000.0) || !(x <= 145.0))
		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 <= -125000.0) || ~((x <= 145.0)))
		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, -125000.0], N[Not[LessEqual[x, 145.0]], $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 < -125000 or 145 < 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%

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

   3. Simplified 99.6%

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

      1. un-div-inv 99.7%

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

      2. add-sqr-sqrt 49.8%

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

      3. sqrt-unprod 41.3%

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

      4. sqr-neg 41.3%

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

      5. sqrt-unprod 0.2%

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

      6. add-sqr-sqrt 0.5%

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

      7. frac-2neg 0.5%

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

      8. div-inv 0.5%

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

      9. fma-def 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 31.9%

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

      12. sqr-neg 31.9%

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

      13. sqrt-unprod 49.6%

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

      14. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 97.5%

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

      1. mul-1-neg 97.5%

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

      2. associate-/l* 97.4%

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

   8. Simplified 97.4%

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

      1. associate-/r/ 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. *-commutative 97.5%

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

      4. *-commutative 97.5%

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

      5. clear-num 97.5%

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

      6. un-div-inv 97.5%

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

      7. quot-tan 97.6%

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

   10. Applied egg-rr 97.6%

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

   if -125000 < x < 145

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 98.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -125000 \lor \neg \left(x \leq 145\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.7) (not (<= x 1.0))) (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.7) || !(x <= 1.0)) {
		tmp = x * (-1.0 / 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 <= (-1.7d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x * ((-1.0d0) / 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 <= -1.7) || !(x <= 1.0)) {
		tmp = x * (-1.0 / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.7) or not (x <= 1.0):
		tmp = x * (-1.0 / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.7) || !(x <= 1.0))
		tmp = Float64(x * Float64(-1.0 / 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 <= -1.7) || ~((x <= 1.0)))
		tmp = x * (-1.0 / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.7], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996 or 1 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

      1. un-div-inv 99.7%

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

      2. add-sqr-sqrt 50.6%

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

      3. sqrt-unprod 42.3%

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

      4. sqr-neg 42.3%

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

      5. sqrt-unprod 0.2%

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

      6. add-sqr-sqrt 0.5%

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

      7. frac-2neg 0.5%

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

      8. div-inv 0.5%

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

      9. fma-def 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 31.4%

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

      12. sqr-neg 31.4%

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

      13. sqrt-unprod 48.8%

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

      14. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 96.3%

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

      1. mul-1-neg 96.3%

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

      2. associate-/l* 96.2%

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

   8. Simplified 96.2%

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

      1. associate-/r/ 96.3%

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

      2. clear-num 96.3%

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

      3. quot-tan 96.3%

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

   10. Applied egg-rr 96.3%

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

   if -1.69999999999999996 < x < 1

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.049:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;B \leq 9:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{B}{\sin B}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B -0.049)
   (/ 1.0 (sin B))
   (if (<= B 9.0)
     (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B))
     (/ (/ B (sin B)) B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= -0.049) {
		tmp = 1.0 / sin(B);
	} else if (B <= 9.0) {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	} else {
		tmp = (B / sin(B)) / 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 <= (-0.049d0)) then
        tmp = 1.0d0 / sin(b)
    else if (b <= 9.0d0) then
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
    else
        tmp = (b / sin(b)) / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= -0.049) {
		tmp = 1.0 / Math.sin(B);
	} else if (B <= 9.0) {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	} else {
		tmp = (B / Math.sin(B)) / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= -0.049:
		tmp = 1.0 / math.sin(B)
	elif B <= 9.0:
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	else:
		tmp = (B / math.sin(B)) / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= -0.049)
		tmp = Float64(1.0 / sin(B));
	elseif (B <= 9.0)
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	else
		tmp = Float64(Float64(B / sin(B)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= -0.049)
		tmp = 1.0 / sin(B);
	elseif (B <= 9.0)
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	else
		tmp = (B / sin(B)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, -0.049], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 9.0], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(B / N[Sin[B], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < -0.049000000000000002

   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%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 49.9%

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

   if -0.049000000000000002 < B < 9

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. div-sub 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around inf 99.4%

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

   if 9 < B

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*r/ 99.6%

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

      4. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 55.2%

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

      1. frac-sub 55.1%

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

      2. associate-/r* 55.2%

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

      3. *-un-lft-identity 55.2%

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

      4. *-commutative 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in B around inf 55.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.049:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;B \leq 9:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{B}{\sin B}}{B}\\ \end{array} \]

Alternative 6: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.5) || !(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 <= (-1.5d0)) .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 <= -1.5) || !(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 <= -1.5) 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 <= -1.5) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.5) || ~((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, -1.5], 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 < -1.5 or 1 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

      1. un-div-inv 99.7%

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

      2. add-sqr-sqrt 50.6%

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

      3. sqrt-unprod 42.3%

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

      4. sqr-neg 42.3%

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

      5. sqrt-unprod 0.2%

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

      6. add-sqr-sqrt 0.5%

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

      7. frac-2neg 0.5%

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

      8. div-inv 0.5%

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

      9. fma-def 0.5%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 31.4%

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

      12. sqr-neg 31.4%

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

      13. sqrt-unprod 48.8%

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

      14. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 96.3%

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

      1. mul-1-neg 96.3%

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

      2. associate-/l* 96.2%

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

   8. Simplified 96.2%

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

      1. associate-/r/ 96.3%

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

      2. *-un-lft-identity 96.3%

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

      3. *-commutative 96.3%

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

      4. *-commutative 96.3%

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

      5. clear-num 96.3%

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

      6. un-div-inv 96.3%

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

      7. quot-tan 96.4%

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

   10. Applied egg-rr 96.4%

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

   if -1.5 < x < 1

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 75.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.0142 \lor \neg \left(B \leq 9\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -0.0142) (not (<= B 9.0)))
   (/ 1.0 (sin B))
   (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -0.0142) || !(B <= 9.0)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((b <= (-0.0142d0)) .or. (.not. (b <= 9.0d0))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -0.0142) || !(B <= 9.0)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -0.0142) or not (B <= 9.0):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -0.0142) || !(B <= 9.0))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -0.0142) || ~((B <= 9.0)))
		tmp = 1.0 / sin(B);
	else
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -0.0142], N[Not[LessEqual[B, 9.0]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -0.014200000000000001 or 9 < B

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 52.2%

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

   if -0.014200000000000001 < B < 9

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. div-sub 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around inf 99.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.0142 \lor \neg \left(B \leq 9\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 8: 51.5% 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. Step-by-step derivation

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 52.1%

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

   1. +-commutative 52.1%

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

   2. mul-1-neg 52.1%

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

   3. sub-neg 52.1%

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

   4. associate--l+ 52.1%

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

   5. *-commutative 52.1%

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

   6. *-commutative 52.1%

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

   7. div-sub 52.1%

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

6. Simplified 52.1%

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

7. Taylor expanded in x around inf 52.4%

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

8. Final simplification 52.4%

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

Alternative 9: 50.0% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 195\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 195.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 195.0)) {
		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 <= 195.0d0))) 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 <= 195.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 195.0):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 195.0))
		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 <= 195.0)))
		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, 195.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 195 < 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%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 49.7%

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

      1. mul-1-neg 49.7%

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

      2. sub-neg 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in x around inf 48.4%

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

      1. mul-1-neg 48.4%

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

      2. distribute-frac-neg 48.4%

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

   9. Simplified 48.4%

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

   if -1 < x < 195

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. sub-neg 54.0%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in x around 0 53.9%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 195\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 10: 51.1% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 52.0%

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

   1. mul-1-neg 52.0%

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

   2. sub-neg 52.0%

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

6. Simplified 52.0%

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

7. Final simplification 52.0%

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

Alternative 11: 26.7% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 52.0%

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

   1. mul-1-neg 52.0%

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

   2. sub-neg 52.0%

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

6. Simplified 52.0%

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

7. Taylor expanded in x around 0 30.9%

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

8. Final simplification 30.9%

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

Reproduce

herbie shell --seed 2023227 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))