VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.7s

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: 99.8% 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. 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. Final simplification 99.8%

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

Alternative 2: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := t\_0 - \frac{1}{\tan B} \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (- t_0 (* (/ 1.0 (tan B)) x))))
   (if (<= t_1 -2e+17)
     (/ (- x) (tan B))
     (if (<= t_1 20.0) t_0 (/ (- 1.0 x) (tan B))))))

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 tmp;
	if (t_1 <= -2e+17) {
		tmp = -x / tan(B);
	} else if (t_1 <= 20.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / tan(B);
	}
	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) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = t_0 - ((1.0d0 / tan(b)) * x)
    if (t_1 <= (-2d+17)) then
        tmp = -x / tan(b)
    else if (t_1 <= 20.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 - x) / tan(b)
    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 tmp;
	if (t_1 <= -2e+17) {
		tmp = -x / Math.tan(B);
	} else if (t_1 <= 20.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = t_0 - ((1.0 / math.tan(B)) * x)
	tmp = 0
	if t_1 <= -2e+17:
		tmp = -x / math.tan(B)
	elif t_1 <= 20.0:
		tmp = t_0
	else:
		tmp = (1.0 - x) / math.tan(B)
	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))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (t_1 <= 20.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / tan(B));
	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);
	tmp = 0.0;
	if (t_1 <= -2e+17)
		tmp = -x / tan(B);
	elseif (t_1 <= 20.0)
		tmp = t_0;
	else
		tmp = (1.0 - x) / tan(B);
	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]}, If[LessEqual[t$95$1, -2e+17], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20.0], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -2e17

   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 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} + \frac{1}{\sin B} \]
   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. frac-add N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 60.3

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

   11. Applied rewrites 60.3%

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

   if -2e17 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 20

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   if 20 < (+.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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} + \frac{1}{\sin B} \]
   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. frac-add N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in B around 0

      \[\leadsto \frac{\color{blue}{1 + -1 \cdot x}}{\tan B} \]
   10. 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 99.2

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

   11. Applied rewrites 99.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} - \frac{1}{\tan B} \cdot x \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;\frac{1}{\sin B} - \frac{1}{\tan B} \cdot x \leq 20:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. *-rgt-identity N/A

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

   10. lower-fma.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 4: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) (tan B))))
   (if (<= x -1.15) t_0 (if (<= x 5.8e-5) (- (/ 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 <= -1.15) {
		tmp = t_0;
	} else if (x <= 5.8e-5) {
		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 <= (-1.15d0)) then
        tmp = t_0
    else if (x <= 5.8d-5) 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 <= -1.15) {
		tmp = t_0;
	} else if (x <= 5.8e-5) {
		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 <= -1.15:
		tmp = t_0
	elif x <= 5.8e-5:
		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 <= -1.15)
		tmp = t_0;
	elseif (x <= 5.8e-5)
		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 <= -1.15)
		tmp = t_0;
	elseif (x <= 5.8e-5)
		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, -1.15], t$95$0, If[LessEqual[x, 5.8e-5], 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 < -1.1499999999999999 or 5.8e-5 < 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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} + \frac{1}{\sin B} \]
   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. frac-add N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in B around 0

      \[\leadsto \frac{\color{blue}{1 + -1 \cdot x}}{\tan B} \]
   10. 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 99.5

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

   11. Applied rewrites 99.5%

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

   if -1.1499999999999999 < x < 5.8e-5

   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

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.15) 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 <= -1.15) {
		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 <= (-1.15d0)) 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 <= -1.15) {
		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 <= -1.15:
		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 <= -1.15)
		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 <= -1.15)
		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, -1.15], 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 < -1.1499999999999999 or 1 < 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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} + \frac{1}{\sin B} \]
   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. frac-add N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 99.1

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

   11. Applied rewrites 99.1%

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

   if -1.1499999999999999 < x < 1

   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 96.3

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

   5. Applied rewrites 96.3%

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

Alternative 6: 63.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, 0.019444444444444445\right) \cdot B, B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.06)
   (/
    (fma
     (fma
      (* (fma 0.022222222222222223 x 0.019444444444444445) B)
      B
      (fma 0.3333333333333333 x 0.16666666666666666))
     (* B B)
     (- 1.0 x))
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.06) {
		tmp = fma(fma((fma(0.022222222222222223, x, 0.019444444444444445) * B), B, fma(0.3333333333333333, x, 0.16666666666666666)), (B * B), (1.0 - x)) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.06)
		tmp = Float64(fma(fma(Float64(fma(0.022222222222222223, x, 0.019444444444444445) * B), B, fma(0.3333333333333333, x, 0.16666666666666666)), Float64(B * B), Float64(1.0 - x)) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.06], N[(N[(N[(N[(N[(0.022222222222222223 * x + 0.019444444444444445), $MachinePrecision] * B), $MachinePrecision] * B + N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.059999999999999998

   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 63.8%

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

   if 0.059999999999999998 < B

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 7: 51.1% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(1.0 / B), $MachinePrecision] + N[(N[(0.3333333333333333 * B), $MachinePrecision] * x + N[((-x) / 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. Taylor expanded in B around 0

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

   1. lower-/.f64 70.7

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

5. Applied rewrites 70.7%

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

6. Taylor expanded in B around 0

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. unpow2 N/A

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

   7. associate-/l* N/A

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

   8. *-rgt-identity N/A

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

   9. associate-*r/ N/A

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

   10. rgt-mult-inverse N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

   14. associate-*r* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. associate-*r/ N/A

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

   18. mul-1-neg N/A

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

   19. lower-/.f64 N/A

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

   20. lower-neg.f64 49.1

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

8. Applied rewrites 49.1%

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

9. Final simplification 49.1%

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

Alternative 8: 51.0% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[(N[(B * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * B + N[(1.0 - x), $MachinePrecision]), $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. 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. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. associate-+r+ N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 48.5

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

7. Applied rewrites 48.5%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 49.0%

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

Alternative 9: 49.7% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.7%

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

      1. rem-exp-log N/A

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

      2. lift-/.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-log.f64 48.1

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 50.8

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

   7. Applied rewrites 50.8%

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

   if -1 < x < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.5%

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

Alternative 10: 50.7% 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 48.8

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

5. Applied rewrites 48.8%

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

Alternative 11: 26.7% 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 48.8

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

5. Applied rewrites 48.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.8%

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

Alternative 12: 3.2% 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. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. associate-+r+ N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 48.5

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

7. Applied rewrites 48.5%

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

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 2.5%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 2.9%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))