VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 10.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. neg-sub0 N/A

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

   3. neg-sub0 N/A

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

   4. un-div-inv N/A

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

   5. distribute-neg-frac N/A

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

   6. tan-quot N/A

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

   7. neg-mul-1 N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. frac-2neg N/A

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

   11. metadata-eval N/A

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

   12. inv-pow N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow-pow N/A

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

   16. pow2 N/A

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

   17. sqr-neg N/A

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

   18. pow-prod-down N/A

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

   19. sqr-pow N/A

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

   20. inv-pow N/A

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

   21. times-frac N/A

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

4. Applied egg-rr 99.7%

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

Alternative 2: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -350:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -350.0) t_0 (if (<= x 1.0) (/ (- 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 <= -350.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		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 <= (-350.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) 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 <= -350.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		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 <= -350.0:
		tmp = t_0
	elif x <= 1.0:
		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 <= -350.0)
		tmp = t_0;
	elseif (x <= 1.0)
		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 <= -350.0)
		tmp = t_0;
	elseif (x <= 1.0)
		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, -350.0], t$95$0, If[LessEqual[x, 1.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -350 or 1 < 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. +-commutative N/A

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

      2. neg-sub0 N/A

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

      3. neg-sub0 N/A

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

      4. un-div-inv N/A

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

      5. distribute-neg-frac N/A

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

      6. tan-quot N/A

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

      7. neg-mul-1 N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. inv-pow N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow-pow N/A

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

      16. pow2 N/A

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

      17. sqr-neg N/A

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

      18. pow-prod-down N/A

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

      19. sqr-pow N/A

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

      20. inv-pow N/A

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

      21. times-frac N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 98.0

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

   7. Simplified 98.0%

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

   if -350 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 98.8

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

   5. Simplified 98.8%

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

      1. frac-add N/A

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

      2. /-rgt-identity N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-rgt-identity N/A

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

      8. remove-double-neg N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. remove-double-neg N/A

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

      11. remove-double-neg N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-neg-out N/A

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

      18. neg-lowering-neg.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 75.6%

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

   8. Taylor expanded in B around 0

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

   10. Simplified 75.6%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. div-sub N/A

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

       5. unsub-neg N/A

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

       6. mul-1-neg N/A

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

       7. /-lowering-/.f64 N/A

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

       8. mul-1-neg N/A

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

       9. unsub-neg N/A

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

       10. --lowering--.f64 N/A

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

       11. sin-lowering-sin.f64 98.9

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

   13. Simplified 98.9%

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

Alternative 3: 62.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.48:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.3333333333333333, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), \mathsf{fma}\left(x, 0.022222222222222223, 0.019444444444444445\right)\right), 0.16666666666666666\right)\right), 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.48)
   (/
    (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 <= 0.48) {
		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 <= 0.48)
		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, 0.48], 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 < 0.47999999999999998

   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

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

   5. Simplified 67.2%

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

   if 0.47999999999999998 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 48.9

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

   5. Simplified 48.9%

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

Alternative 4: 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

3. Taylor expanded in B around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 74.3

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

5. Simplified 74.3%

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

   1. frac-add N/A

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

   2. /-rgt-identity N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-rgt-identity N/A

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

   5. *-rgt-identity N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. /-rgt-identity N/A

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

   8. remove-double-neg N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. remove-double-neg N/A

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

   11. remove-double-neg N/A

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

   12. neg-lowering-neg.f64 N/A

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

   13. remove-double-neg N/A

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

   14. neg-mul-1 N/A

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

   15. remove-double-neg N/A

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

   16. mul-1-neg N/A

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

   17. distribute-lft-neg-out N/A

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

   18. neg-lowering-neg.f64 N/A

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

   19. *-lowering-*.f64 N/A

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

7. Applied egg-rr 59.0%

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

8. Taylor expanded in B around 0

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

10. Simplified 59.6%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

    4. div-sub N/A

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

    5. unsub-neg N/A

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

    6. mul-1-neg N/A

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

    7. /-lowering-/.f64 N/A

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

    8. mul-1-neg N/A

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

    9. unsub-neg N/A

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

    10. --lowering--.f64 N/A

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

    11. sin-lowering-sin.f64 75.3

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

13. Simplified 75.3%

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

Alternative 5: 50.7% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. /-lowering-/.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. --lowering--.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. accelerator-lowering-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 50.8

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

5. Simplified 50.8%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \color{blue}{\frac{1}{3} \cdot x}, 1\right) - x}{B} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-lowering-*.f64 50.8

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

8. Simplified 50.8%

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

Alternative 6: 49.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 54000:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ x (- B))))
   (if (<= x -1.0) t_0 (if (<= x 54000.0) (/ 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 <= 54000.0) {
		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 <= 54000.0d0) 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 <= 54000.0) {
		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 <= 54000.0:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(x / Float64(-B))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 54000.0)
		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 <= 54000.0)
		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, 54000.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 54000 < 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. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 45.2

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

   5. Simplified 45.2%

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

   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. neg-lowering-neg.f64 43.5

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

   8. Simplified 43.5%

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

   if -1 < x < 54000

   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. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 54.3

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

   8. Simplified 54.3%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;x \leq 54000:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 74.3

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

5. Simplified 74.3%

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

6. Taylor expanded in B around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. --lowering--.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 50.6

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

8. Simplified 50.6%

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

Alternative 8: 50.6% 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. /-lowering-/.f64 N/A

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

   2. --lowering--.f64 50.4

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

5. Simplified 50.4%

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

Alternative 9: 26.2% 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 B around 0

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

   1. /-lowering-/.f64 N/A

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

   2. --lowering--.f64 50.4

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

5. Simplified 50.4%

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

6. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 31.1

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

8. Simplified 31.1%

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

Alternative 10: 3.1% accurate, 38.8× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(B, x):
	return B * 0.16666666666666666

Julia

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

MATLAB

function tmp = code(B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $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

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

   1. /-lowering-/.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. --lowering--.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. accelerator-lowering-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 50.8

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

5. Simplified 50.8%

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

6. Taylor expanded in B around inf

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

   1. *-lowering-*.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. accelerator-lowering-fma.f64 2.9

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

8. Simplified 2.9%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{B \cdot \frac{1}{6}} \]

    2. *-lowering-*.f64 3.1

       \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

11. Simplified 3.1%

    \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))