VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 9.9s

Alternatives: 12

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: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

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

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 99.7

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

   8. lower-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-tan.f64 N/A

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

   16. tan-quot N/A

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

   17. lift-sin.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. clear-num N/A

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

   20. associate-/l* N/A

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

   21. lift-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := t\_0 - \frac{1}{\tan B} \cdot x\\ t_2 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;t\_1 \leq -200000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1 (- t_0 (* (/ 1.0 (tan B)) x)))
        (t_2 (/ (- 1.0 x) (tan B))))
   (if (<= t_1 -200000000000.0) t_2 (if (<= t_1 1000.0) t_0 t_2))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = t_0 - ((1.0 / tan(B)) * x);
	double t_2 = (1.0 - x) / tan(B);
	double tmp;
	if (t_1 <= -200000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = t_0 - ((1.0d0 / tan(b)) * x)
    t_2 = (1.0d0 - x) / tan(b)
    if (t_1 <= (-200000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 1000.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = t_0 - ((1.0 / Math.tan(B)) * x);
	double t_2 = (1.0 - x) / Math.tan(B);
	double tmp;
	if (t_1 <= -200000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = t_0 - ((1.0 / math.tan(B)) * x)
	t_2 = (1.0 - x) / math.tan(B)
	tmp = 0
	if t_1 <= -200000000000.0:
		tmp = t_2
	elif t_1 <= 1000.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(t_0 - Float64(Float64(1.0 / tan(B)) * x))
	t_2 = Float64(Float64(1.0 - x) / tan(B))
	tmp = 0.0
	if (t_1 <= -200000000000.0)
		tmp = t_2;
	elseif (t_1 <= 1000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 1.0 / sin(B);
	t_1 = t_0 - ((1.0 / tan(B)) * x);
	t_2 = (1.0 - x) / tan(B);
	tmp = 0.0;
	if (t_1 <= -200000000000.0)
		tmp = t_2;
	elseif (t_1 <= 1000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200000000000.0], t$95$2, If[LessEqual[t$95$1, 1000.0], t$95$0, t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -2e11 or 1e3 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-add N/A

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

      6. /-rgt-identity N/A

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

      7. *-rgt-identity N/A

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

      8. lift-fma.f64 N/A

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

      9. /-rgt-identity N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if -2e11 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 1e3

   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 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} - \frac{1}{\tan B} \cdot x \leq -200000000000:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{elif}\;\frac{1}{\sin B} - \frac{1}{\tan B} \cdot x \leq 1000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;x \leq -8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) (tan B))))
   (if (<= x -8.0) t_0 (if (<= x 2.0) (- (/ 1.0 (sin B)) (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / tan(B);
	double tmp;
	if (x <= -8.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		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 - x) / tan(b)
    if (x <= (-8.0d0)) then
        tmp = t_0
    else if (x <= 2.0d0) 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 - x) / Math.tan(B);
	double tmp;
	if (x <= -8.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / math.tan(B)
	tmp = 0
	if x <= -8.0:
		tmp = t_0
	elif x <= 2.0:
		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 - x) / tan(B))
	tmp = 0.0
	if (x <= -8.0)
		tmp = t_0;
	elseif (x <= 2.0)
		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 - x) / tan(B);
	tmp = 0.0;
	if (x <= -8.0)
		tmp = t_0;
	elseif (x <= 2.0)
		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 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.0], t$95$0, If[LessEqual[x, 2.0], 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 < -8 or 2 < x

   1. Initial program 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-add N/A

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

      6. /-rgt-identity N/A

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

      7. *-rgt-identity N/A

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

      8. lift-fma.f64 N/A

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

      9. /-rgt-identity N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.7

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 99.0

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

   9. Applied rewrites 99.0%

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

   if -8 < x < 2

   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

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -96:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{+14}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -96.0) t_0 (if (<= x 1e+14) (/ (- 1.0 x) (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -96.0) {
		tmp = t_0;
	} else if (x <= 1e+14) {
		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 = -x / tan(b)
    if (x <= (-96.0d0)) then
        tmp = t_0
    else if (x <= 1d+14) 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 = -x / Math.tan(B);
	double tmp;
	if (x <= -96.0) {
		tmp = t_0;
	} else if (x <= 1e+14) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if x <= -96.0:
		tmp = t_0
	elif x <= 1e+14:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -96.0)
		tmp = t_0;
	elseif (x <= 1e+14)
		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 = -x / tan(B);
	tmp = 0.0;
	if (x <= -96.0)
		tmp = t_0;
	elseif (x <= 1e+14)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -96.0], t$95$0, If[LessEqual[x, 1e+14], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -96 or 1e14 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 98.8

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

   7. Applied rewrites 98.8%

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

   9. Applied rewrites 98.9%

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

   if -96 < x < 1e14

   1. Initial program 99.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 99.8

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-tan.f64 N/A

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

      16. tan-quot N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. clear-num N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   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 97.9

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

   9. Applied rewrites 97.9%

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

Alternative 5: 97.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / 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 = -x / tan(b)
    if (x <= (-1.45d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 96.9

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

   7. Applied rewrites 96.9%

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

   9. Applied rewrites 97.0%

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

   if -1.44999999999999996 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 97.0

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

   5. Applied rewrites 97.0%

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

Alternative 6: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.00011:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.00011) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.00011) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.00011d0) then
        tmp = (1.0d0 - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 0.00011) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.00011:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.00011)
		tmp = Float64(Float64(1.0 - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 0.00011)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.00011], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.10000000000000004e-4

   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 70.6

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

   5. Applied rewrites 70.6%

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

   if 1.10000000000000004e-4 < 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. lower-/.f64 N/A

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

      2. lower-sin.f64 47.3

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

   5. Applied rewrites 47.3%

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

Alternative 7: 51.7% 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 99.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-out-- N/A

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

   7. associate--l+ N/A

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

   8. associate--l+ N/A

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

7. Applied rewrites 53.5%

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

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 53.6%

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

Alternative 8: 51.6% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $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. 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 53.5

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

5. Applied rewrites 53.5%

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

Alternative 9: 50.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -1.0) t_0 (if (<= x 0.65) (/ 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 <= 0.65) {
		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 <= 0.65d0) 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 <= 0.65) {
		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 <= 0.65:
		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 <= 0.65)
		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 <= 0.65)
		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, 0.65], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.650000000000000022 < x

   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 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 52.5%

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

   if -1 < x < 0.650000000000000022

   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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.1%

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

Alternative 10: 51.5% 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 53.2

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

5. Applied rewrites 53.2%

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

Alternative 11: 26.4% 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. lower-/.f64 N/A

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

   2. lower--.f64 53.2

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

5. Applied rewrites 53.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.1%

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

Alternative 12: 3.1% accurate, 38.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(0.16666666666666666 * B), $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. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-out-- N/A

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

   7. associate--l+ N/A

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

   8. associate--l+ N/A

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

7. Applied rewrites 53.5%

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

8. Taylor expanded in B around inf

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

10. Applied rewrites 3.0%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 3.3%

    \[\leadsto 0.16666666666666666 \cdot B \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))