VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.8%

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

   10. *-rgt-identity 99.8%

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

   11. tan-neg 99.8%

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

   12. distribute-neg-frac2 99.8%

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

   13. distribute-neg-frac 99.8%

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

   14. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-11} \lor \neg \left(x \leq 6.5 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.2e-11) (not (<= x 6.5e-15)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.2e-11) || !(x <= 6.5e-15)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.2d-11)) .or. (.not. (x <= 6.5d-15))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -3.2e-11) || !(x <= 6.5e-15)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.2e-11) or not (x <= 6.5e-15):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3.2e-11) || !(x <= 6.5e-15))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -3.2e-11) || ~((x <= 6.5e-15)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.2e-11], N[Not[LessEqual[x, 6.5e-15]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999994e-11 or 6.49999999999999991e-15 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-frac-neg2 99.6%

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

      6. tan-neg 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 99.0%

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

   if -3.19999999999999994e-11 < x < 6.49999999999999991e-15

   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 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-11} \lor \neg \left(x \leq 6.5 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-frac-neg2 99.6%

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

      6. tan-neg 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

      1. add-exp-log 53.3%

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

      2. log-rec 53.3%

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

   6. Applied egg-rr 53.3%

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

      1. exp-neg 53.3%

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

      2. add-exp-log 99.8%

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

      3. add-sqr-sqrt 53.3%

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

      4. associate-/r* 53.3%

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

      5. metadata-eval 53.3%

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

      6. sqrt-div 53.3%

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

      7. inv-pow 53.3%

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

      8. pow-to-exp 53.3%

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

      9. *-commutative 53.3%

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

      10. neg-mul-1 53.3%

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

      11. add-sqr-sqrt 53.3%

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

      12. sqrt-unprod 53.3%

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

      13. sqr-neg 53.3%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 51.9%

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

      16. add-exp-log 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. *-inverses 95.7%

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

   10. Simplified 95.7%

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

   if -1.44999999999999996 < 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 98.5%

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

4. Final simplification 97.1%

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

Alternative 4: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 240:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 240.0)
   (+
    (* B 0.16666666666666666)
    (+ (/ (- 1.0 x) B) (* B (* x 0.3333333333333333))))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 240.0) {
		tmp = (B * 0.16666666666666666) + (((1.0 - x) / B) + (B * (x * 0.3333333333333333)));
	} 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 <= 240.0d0) then
        tmp = (b * 0.16666666666666666d0) + (((1.0d0 - x) / b) + (b * (x * 0.3333333333333333d0)))
    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 <= 240.0) {
		tmp = (B * 0.16666666666666666) + (((1.0 - x) / B) + (B * (x * 0.3333333333333333)));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 240.0:
		tmp = (B * 0.16666666666666666) + (((1.0 - x) / B) + (B * (x * 0.3333333333333333)))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 240.0)
		tmp = Float64(Float64(B * 0.16666666666666666) + Float64(Float64(Float64(1.0 - x) / B) + Float64(B * Float64(x * 0.3333333333333333))));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 240.0)
		tmp = (B * 0.16666666666666666) + (((1.0 - x) / B) + (B * (x * 0.3333333333333333)));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 240.0], N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 240

   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 67.7%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   4. Taylor expanded in x around 0 67.6%

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

      1. +-commutative 67.6%

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

      2. *-un-lft-identity 67.6%

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

      3. fma-define 67.6%

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

      4. *-commutative 67.6%

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

      5. fma-neg 67.6%

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

      6. distribute-neg-frac 67.6%

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

      7. metadata-eval 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. fma-undefine 67.6%

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

      2. *-lft-identity 67.6%

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

      3. fma-undefine 67.6%

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

      4. distribute-rgt-out 67.7%

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

      5. associate-*l* 67.7%

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

      6. +-commutative 67.7%

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

      7. associate-+r+ 67.7%

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

      8. associate-*l/ 67.7%

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

      9. associate-*r/ 67.7%

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

      10. mul-1-neg 67.7%

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

      11. sub-neg 67.7%

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

      12. div-sub 67.8%

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

      13. *-commutative 67.8%

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

   8. Simplified 67.8%

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

   if 240 < 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 51.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 240:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.0% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 55.1%

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

4. Taylor expanded in x around 0 55.1%

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

   1. +-commutative 55.1%

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

   2. *-un-lft-identity 55.1%

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

   3. fma-define 55.1%

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

   4. *-commutative 55.1%

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

   5. fma-neg 55.1%

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

   6. distribute-neg-frac 55.1%

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

   7. metadata-eval 55.1%

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

6. Applied egg-rr 55.1%

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

   1. fma-undefine 55.1%

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

   2. *-lft-identity 55.1%

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

   3. fma-undefine 55.1%

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

   4. distribute-rgt-out 55.1%

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

   5. associate-*l* 55.1%

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

   6. +-commutative 55.1%

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

   7. associate-+r+ 55.1%

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

   8. associate-*l/ 55.1%

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

   9. associate-*r/ 55.1%

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

   10. mul-1-neg 55.1%

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

   11. sub-neg 55.1%

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

   12. div-sub 55.2%

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

   13. *-commutative 55.2%

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

8. Simplified 55.2%

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

9. Final simplification 55.2%

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

Alternative 6: 50.9% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 54.0%

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

   4. Taylor expanded in x around inf 50.7%

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

      1. associate-*r/ 50.7%

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

      2. neg-mul-1 50.7%

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

   6. Simplified 50.7%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 55.6%

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

   4. Taylor expanded in x around 0 54.3%

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

4. Final simplification 52.5%

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

Alternative 7: 52.0% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 55.1%

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

4. Taylor expanded in x around 0 55.1%

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

5. Taylor expanded in B around 0 55.0%

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

6. Final simplification 55.0%

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

Alternative 8: 51.8% 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 54.8%

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

4. Final simplification 54.8%

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

Alternative 9: 27.1% 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 54.8%

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

4. Taylor expanded in x around 0 28.6%

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

5. Final simplification 28.6%

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

Reproduce

herbie shell --seed 2024067 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))