VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 5.4s

Alternatives: 11

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 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;\frac{1}{B} - \frac{1}{\tan B} \cdot x\\ \mathbf{elif}\;x \leq 72000000000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -2.7)
   (- (/ 1.0 B) (* (/ 1.0 (tan B)) x))
   (if (<= x 72000000000000.0)
     (- (/ 1.0 (sin B)) (/ x B))
     (/ (* (- x) (cos B)) (sin B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = (1.0 / B) - ((1.0 / tan(B)) * x);
	} else if (x <= 72000000000000.0) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (-x * cos(B)) / 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 <= (-2.7d0)) then
        tmp = (1.0d0 / b) - ((1.0d0 / tan(b)) * x)
    else if (x <= 72000000000000.0d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (-x * cos(b)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = (1.0 / B) - ((1.0 / Math.tan(B)) * x);
	} else if (x <= 72000000000000.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -2.7:
		tmp = (1.0 / B) - ((1.0 / math.tan(B)) * x)
	elif x <= 72000000000000.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (-x * math.cos(B)) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -2.7)
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(1.0 / tan(B)) * x));
	elseif (x <= 72000000000000.0)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -2.7)
		tmp = (1.0 / B) - ((1.0 / tan(B)) * x);
	elseif (x <= 72000000000000.0)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (-x * cos(B)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -2.7], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 72000000000000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7000000000000002

   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-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 51.2

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left({\sin B}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 51.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 97.9

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

   7. Applied rewrites 97.9%

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

   if -2.7000000000000002 < x < 7.2e13

   1. Initial program 99.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 7.2e13 < x

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lift-tan.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 99.5

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

   9. Applied rewrites 99.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;\frac{1}{B} - \frac{1}{\tan B} \cdot x\\ \mathbf{elif}\;x \leq 72000000000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% 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 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. lift-tan.f64 N/A

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

   14. tan-quot N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. associate-/r/ N/A

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

   18. associate-*l/ N/A

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

   19. sub-div N/A

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

   20. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

Alternative 4: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{1}{\tan B} \cdot x\\ \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (* (/ 1.0 (tan B)) x))))
   (if (<= x -2.7) t_0 (if (<= x 3.5e-7) (- (/ 1.0 (sin B)) (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - ((1.0 / tan(B)) * x);
	double tmp;
	if (x <= -2.7) {
		tmp = t_0;
	} else if (x <= 3.5e-7) {
		tmp = (1.0 / sin(B)) - (x / 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) - ((1.0d0 / tan(b)) * x)
    if (x <= (-2.7d0)) then
        tmp = t_0
    else if (x <= 3.5d-7) then
        tmp = (1.0d0 / sin(b)) - (x / 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) - ((1.0 / Math.tan(B)) * x);
	double tmp;
	if (x <= -2.7) {
		tmp = t_0;
	} else if (x <= 3.5e-7) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - ((1.0 / math.tan(B)) * x)
	tmp = 0
	if x <= -2.7:
		tmp = t_0
	elif x <= 3.5e-7:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(Float64(1.0 / tan(B)) * x))
	tmp = 0.0
	if (x <= -2.7)
		tmp = t_0;
	elseif (x <= 3.5e-7)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 / B) - ((1.0 / tan(B)) * x);
	tmp = 0.0;
	if (x <= -2.7)
		tmp = t_0;
	elseif (x <= 3.5e-7)
		tmp = (1.0 / sin(B)) - (x / 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[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7], t$95$0, If[LessEqual[x, 3.5e-7], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002 or 3.49999999999999984e-7 < x

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 47.4

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left({\sin B}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 47.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if -2.7000000000000002 < x < 3.49999999999999984e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;\frac{1}{B} - \frac{1}{\tan B} \cdot x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{1}{\tan B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.72:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, 0.019444444444444445\right) \cdot B, B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.72)
   (/
    (fma
     (fma
      (* (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 <= 0.72) {
		tmp = fma(fma((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 <= 0.72)
		tmp = Float64(fma(fma(Float64(fma(0.022222222222222223, x, 0.019444444444444445) * 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, 0.72], N[(N[(N[(N[(N[(0.022222222222222223 * x + 0.019444444444444445), $MachinePrecision] * B), $MachinePrecision] * B + 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 < 0.71999999999999997

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

   7. Applied rewrites 67.7%

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

   if 0.71999999999999997 < B

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 51.7

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

   9. Applied rewrites 51.7%

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

Alternative 6: 76.8% 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 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. lift-tan.f64 N/A

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

   14. tan-quot N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. associate-/r/ N/A

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

   18. associate-*l/ N/A

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

   19. sub-div N/A

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

   20. lower-/.f64 N/A

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

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 76.7

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

9. Applied rewrites 76.7%

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

Alternative 7: 52.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}}{B} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right)} - x}{B} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{3} \cdot x\right) \cdot {B}^{2}} + 1\right) - x}{B} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{3} \cdot x, {B}^{2}, 1\right)} - x}{B} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot x + \frac{1}{6}}, {B}^{2}, 1\right) - x}{B} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right)}, {B}^{2}, 1\right) - x}{B} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right), \color{blue}{B \cdot B}, 1\right) - x}{B} \]

   9. lower-*.f64 51.8

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

7. Applied rewrites 51.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{6} \cdot B + \color{blue}{\left(x \cdot \left(\frac{1}{3} \cdot B - \frac{1}{B}\right) + \frac{1}{B}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 51.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, B, \frac{-1}{B}\right), \color{blue}{x}, \mathsf{fma}\left(0.16666666666666666, B, \frac{1}{B}\right)\right) \]
11. Add Preprocessing

Alternative 8: 52.0% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 51.7

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

5. Applied rewrites 51.7%

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

7. Applied rewrites 37.0%

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

9. Applied rewrites 51.8%

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

10. Final simplification 51.8%

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

Alternative 9: 50.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -3.2e-7) t_0 (if (<= x 1.0) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -3.2e-7) {
		tmp = t_0;
	} else if (x <= 1.0) {
		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 <= (-3.2d-7)) then
        tmp = t_0
    else if (x <= 1.0d0) 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 <= -3.2e-7) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -3.2e-7:
		tmp = t_0
	elif x <= 1.0:
		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 <= -3.2e-7)
		tmp = t_0;
	elseif (x <= 1.0)
		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 <= -3.2e-7)
		tmp = t_0;
	elseif (x <= 1.0)
		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, -3.2e-7], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000001e-7 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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 50.5

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.3%

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

   if -3.2000000000000001e-7 < x < 1

   1. Initial program 99.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.4%

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

Alternative 10: 51.9% 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 99.8%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 51.7

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

5. Applied rewrites 51.7%

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

Alternative 11: 26.9% 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 99.8%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 51.7

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

5. Applied rewrites 51.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.5%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))