VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.6s

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 99.7%

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around inf

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

   1. associate-/l* N/A

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

   2. rgt-mult-inverse N/A

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

   3. associate-*r/ N/A

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

   4. associate-/r* N/A

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

   5. distribute-lft-out-- N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. neg-sub0 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} + \frac{1}{x \cdot \sin B}\right) \]

   9. associate-+l- N/A

      \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{\cos B}{\sin B} - \frac{1}{x \cdot \sin B}\right)\right)} \]

   10. neg-sub0 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

7. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. metadata-eval N/A

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

   8. lift-neg.f64 N/A

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

   9. frac-2neg N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-lft-in N/A

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

   15. neg-mul-1 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. distribute-rgt-neg-in N/A

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

   19. metadata-eval N/A

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

   20. lower-fma.f64 N/A

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

   21. lower-neg.f64 99.8

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

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B}} \]
10. Add Preprocessing

Alternative 3: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -0.00035:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 720000000000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -0.00035)
     t_0
     (if (<= x 720000000000.0) (/ (- 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 <= -0.00035) {
		tmp = t_0;
	} else if (x <= 720000000000.0) {
		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 <= (-0.00035d0)) then
        tmp = t_0
    else if (x <= 720000000000.0d0) 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 <= -0.00035) {
		tmp = t_0;
	} else if (x <= 720000000000.0) {
		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 <= -0.00035:
		tmp = t_0
	elif x <= 720000000000.0:
		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 <= -0.00035)
		tmp = t_0;
	elseif (x <= 720000000000.0)
		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 <= -0.00035)
		tmp = t_0;
	elseif (x <= 720000000000.0)
		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, -0.00035], t$95$0, If[LessEqual[x, 720000000000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999996e-4 or 7.2e11 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -3.49999999999999996e-4 < x < 7.2e11

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around inf

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

      1. associate-/l* N/A

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

      2. rgt-mult-inverse N/A

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

      3. associate-*r/ N/A

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

      4. associate-/r* N/A

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

      5. distribute-lft-out-- N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} + \frac{1}{x \cdot \sin B}\right) \]

      9. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{\cos B}{\sin B} - \frac{1}{x \cdot \sin B}\right)\right)} \]

      10. neg-sub0 N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

   7. Applied rewrites 99.8%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. neg-mul-1 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. distribute-rgt-neg-in N/A

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

      19. metadata-eval N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-neg.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in B around 0

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

       1. mul-1-neg N/A

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

       2. sub-neg N/A

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

       3. lower--.f64 99.8

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

   12. Applied rewrites 99.8%

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

Alternative 4: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.84:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.3333333333333333, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), \mathsf{fma}\left(x, 0.022222222222222223, 0.019444444444444445\right)\right), 0.16666666666666666\right)\right), 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.84)
   (/
    (fma
     (* B B)
     (fma
      x
      0.3333333333333333
      (fma
       (* B B)
       (fma
        B
        (* B (fma x 0.0021164021164021165 0.00205026455026455))
        (fma x 0.022222222222222223 0.019444444444444445))
       0.16666666666666666))
     (- 1.0 x))
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.84) {
		tmp = fma((B * B), fma(x, 0.3333333333333333, fma((B * B), fma(B, (B * fma(x, 0.0021164021164021165, 0.00205026455026455)), fma(x, 0.022222222222222223, 0.019444444444444445)), 0.16666666666666666)), (1.0 - x)) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.84)
		tmp = Float64(fma(Float64(B * B), fma(x, 0.3333333333333333, fma(Float64(B * B), fma(B, Float64(B * fma(x, 0.0021164021164021165, 0.00205026455026455)), fma(x, 0.022222222222222223, 0.019444444444444445)), 0.16666666666666666)), Float64(1.0 - x)) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.84], N[(N[(N[(B * B), $MachinePrecision] * N[(x * 0.3333333333333333 + N[(N[(B * B), $MachinePrecision] * N[(B * N[(B * N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision]), $MachinePrecision] + N[(x * 0.022222222222222223 + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $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 < 0.839999999999999969

   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

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

   5. Applied rewrites 67.9%

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

   if 0.839999999999999969 < 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 47.4

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

   5. Applied rewrites 47.4%

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

Alternative 5: 76.6% 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 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around inf

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

   1. associate-/l* N/A

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

   2. rgt-mult-inverse N/A

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

   3. associate-*r/ N/A

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

   4. associate-/r* N/A

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

   5. distribute-lft-out-- N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. neg-sub0 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} + \frac{1}{x \cdot \sin B}\right) \]

   9. associate-+l- N/A

      \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{\cos B}{\sin B} - \frac{1}{x \cdot \sin B}\right)\right)} \]

   10. neg-sub0 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

7. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. metadata-eval N/A

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

   8. lift-neg.f64 N/A

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

   9. frac-2neg N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-lft-in N/A

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

   15. neg-mul-1 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. distribute-rgt-neg-in N/A

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

   19. metadata-eval N/A

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

   20. lower-fma.f64 N/A

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

   21. lower-neg.f64 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in B around 0

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

    1. mul-1-neg N/A

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

    2. sub-neg N/A

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

    3. lower--.f64 77.1

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

12. Applied rewrites 77.1%

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

Alternative 6: 51.3% 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 51.6

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

5. Applied rewrites 51.6%

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

   1. div-sub N/A

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

   2. frac-sub N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 40.6

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

7. Applied rewrites 40.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 51.7

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

   7. lift-*.f64 N/A

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

   8. *-lft-identity 51.7

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

9. Applied rewrites 51.7%

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

Alternative 7: 50.1% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ x B)))) (if (<= x -1.0) 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 <= -1.0) {
		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 <= (-1.0d0)) 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 <= -1.0) {
		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 <= -1.0:
		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 <= -1.0)
		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 <= -1.0)
		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, -1.0], 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 < -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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.0

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

   8. Applied rewrites 54.0%

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

   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 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 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 46.4

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

   8. Applied rewrites 46.4%

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

4. Final simplification 50.5%

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

Alternative 8: 51.3% 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 51.6

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

5. Applied rewrites 51.6%

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

Alternative 9: 26.8% 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 47.1

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

5. Applied rewrites 47.1%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 22.9

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

8. Applied rewrites 22.9%

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

Alternative 10: 2.9% accurate, 19.4× speedup

Math

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

FPCore

(FPCore (B x)
 :precision binary64
 (* B (fma x 0.3333333333333333 0.16666666666666666)))

C

double code(double B, double x) {
	return B * fma(x, 0.3333333333333333, 0.16666666666666666);
}

Julia

function code(B, x)
	return Float64(B * fma(x, 0.3333333333333333, 0.16666666666666666))
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around inf

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

   1. associate-/l* N/A

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

   2. rgt-mult-inverse N/A

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

   3. associate-*r/ N/A

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

   4. associate-/r* N/A

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

   5. distribute-lft-out-- N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. neg-sub0 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} + \frac{1}{x \cdot \sin B}\right) \]

   9. associate-+l- N/A

      \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{\cos B}{\sin B} - \frac{1}{x \cdot \sin B}\right)\right)} \]

   10. neg-sub0 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 51.3%

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

11. Taylor expanded in B around inf

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f64 2.7

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

13. Applied rewrites 2.7%

    \[\leadsto \color{blue}{B \cdot \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))