VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.2s

Alternatives: 10

Speedup: 1.1×

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 99.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 3500000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.45)
   (- (/ 1.0 B) (/ x (tan B)))
   (if (<= x 3500000.0) (/ (- 1.0 x) (sin B)) (* (/ (- x) (sin B)) (cos B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (x <= 3500000.0) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = (-x / sin(B)) * cos(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)) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (x <= 3500000.0d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = (-x / sin(b)) * cos(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (x <= 3500000.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = (-x / Math.sin(B)) * Math.cos(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.45:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif x <= 3500000.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = (-x / math.sin(B)) * math.cos(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.45)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (x <= 3500000.0)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.45)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (x <= 3500000.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = (-x / sin(B)) * cos(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.45], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3500000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996

   1. Initial program 99.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

   if -1.44999999999999996 < x < 3.5e6

   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. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 99.7

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. lift-tan.f64 N/A

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

      17. tan-quot N/A

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

      18. lift-sin.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. clear-num N/A

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

      21. associate-*l/ N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if 3.5e6 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 3500000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 99.6

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

   8. lower-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-/.f64 N/A

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

   16. lift-tan.f64 N/A

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

   17. tan-quot N/A

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

   18. lift-sin.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. clear-num N/A

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

   21. associate-*l/ N/A

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

6. Applied rewrites 99.7%

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

Alternative 4: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -1.45) t_0 (if (<= x 5.8e-13) (/ (- 1.0 x) (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 5.8e-13) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-1.45d0)) then
        tmp = t_0
    else if (x <= 5.8d-13) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 5.8e-13) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -1.45:
		tmp = t_0
	elif x <= 5.8e-13:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.45)
		tmp = t_0;
	elseif (x <= 5.8e-13)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -1.45)
		tmp = t_0;
	elseif (x <= 5.8e-13)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45], t$95$0, If[LessEqual[x, 5.8e-13], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 5.7999999999999995e-13 < x

   1. Initial program 99.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if -1.44999999999999996 < x < 5.7999999999999995e-13

   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. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 99.8

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. lift-tan.f64 N/A

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

      17. tan-quot N/A

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

      18. lift-sin.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. clear-num N/A

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

      21. associate-*l/ N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 99.1

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

   9. Applied rewrites 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right) \cdot B, B, \mathsf{fma}\left(0.022222222222222223, x, 0.019444444444444445\right)\right), B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.8)
   (/
    (fma
     (fma
      (fma
       (* (fma x 0.0021164021164021165 0.00205026455026455) B)
       B
       (fma 0.022222222222222223 x 0.019444444444444445))
      (* B B)
      (fma 0.3333333333333333 x 0.16666666666666666))
     (* B B)
     (- 1.0 x))
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.8) {
		tmp = fma(fma(fma((fma(x, 0.0021164021164021165, 0.00205026455026455) * B), B, fma(0.022222222222222223, x, 0.019444444444444445)), (B * B), fma(0.3333333333333333, x, 0.16666666666666666)), (B * B), (1.0 - x)) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.8)
		tmp = Float64(fma(fma(fma(Float64(fma(x, 0.0021164021164021165, 0.00205026455026455) * B), B, fma(0.022222222222222223, x, 0.019444444444444445)), Float64(B * B), fma(0.3333333333333333, x, 0.16666666666666666)), Float64(B * B), Float64(1.0 - x)) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.8], N[(N[(N[(N[(N[(N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision] * B), $MachinePrecision] * B + N[(0.022222222222222223 * x + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.80000000000000004

   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. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{31}{15120} + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right)}{B}} \]

   7. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right) \cdot B, B, \mathsf{fma}\left(0.022222222222222223, x, 0.019444444444444445\right)\right), B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}} \]

   if 0.80000000000000004 < B

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 48.4

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

   5. Applied rewrites 48.4%

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

Alternative 6: 76.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 99.6

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

   8. lower-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-/.f64 N/A

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

   16. lift-tan.f64 N/A

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

   17. tan-quot N/A

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

   18. lift-sin.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. clear-num N/A

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

   21. associate-*l/ N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in B around 0

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

   1. lower--.f64 73.0

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

9. Applied rewrites 73.0%

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

Alternative 7: 50.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 45.3

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

5. Applied rewrites 45.3%

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

7. Applied rewrites 31.9%

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

9. Applied rewrites 45.5%

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

10. Final simplification 45.5%

    \[\leadsto \frac{\frac{B - B \cdot x}{B}}{B} \]
11. Add Preprocessing

Alternative 8: 49.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -1.0) t_0 (if (<= x 9e+21) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 9e+21) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / b
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 9d+21) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 9e+21) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 9e+21:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 9e+21)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 9e+21)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 9e+21], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 9e21 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 44.5

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

   5. Applied rewrites 44.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.2%

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

   if -1 < x < 9e21

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 46.2

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.6%

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

Alternative 9: 50.8% accurate, 15.5× 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 99.6%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 45.3

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

5. Applied rewrites 45.3%

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

Alternative 10: 27.1% accurate, 19.4× 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 99.6%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 45.3

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

5. Applied rewrites 45.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
7. Step-by-step derivation

8. Applied rewrites 23.7%

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

Reproduce

herbie shell --seed 2024257 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))