VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.7s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. un-div-inv N/A

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

   8. neg-mul-1 N/A

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

   9. lift-tan.f64 N/A

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

   10. tan-quot N/A

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

   11. lift-sin.f64 N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ t_2 := t\_0 + t\_1\\ t_3 := t\_1 + \frac{1}{B}\\ \mathbf{if}\;t\_2 \leq -100000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 400000:\\ \;\;\;\;t\_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1 (* x (/ -1.0 (tan B))))
        (t_2 (+ t_0 t_1))
        (t_3 (+ t_1 (/ 1.0 B))))
   (if (<= t_2 -100000000000.0)
     t_3
     (if (<= t_2 400000.0) (- t_0 (/ x B)) t_3))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x * (-1.0 / tan(B));
	double t_2 = t_0 + t_1;
	double t_3 = t_1 + (1.0 / B);
	double tmp;
	if (t_2 <= -100000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 400000.0) {
		tmp = t_0 - (x / B);
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x * ((-1.0d0) / tan(b))
    t_2 = t_0 + t_1
    t_3 = t_1 + (1.0d0 / b)
    if (t_2 <= (-100000000000.0d0)) then
        tmp = t_3
    else if (t_2 <= 400000.0d0) then
        tmp = t_0 - (x / b)
    else
        tmp = t_3
    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 = x * (-1.0 / Math.tan(B));
	double t_2 = t_0 + t_1;
	double t_3 = t_1 + (1.0 / B);
	double tmp;
	if (t_2 <= -100000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 400000.0) {
		tmp = t_0 - (x / B);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x * (-1.0 / math.tan(B))
	t_2 = t_0 + t_1
	t_3 = t_1 + (1.0 / B)
	tmp = 0
	if t_2 <= -100000000000.0:
		tmp = t_3
	elif t_2 <= 400000.0:
		tmp = t_0 - (x / B)
	else:
		tmp = t_3
	return tmp

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	t_2 = Float64(t_0 + t_1)
	t_3 = Float64(t_1 + Float64(1.0 / B))
	tmp = 0.0
	if (t_2 <= -100000000000.0)
		tmp = t_3;
	elseif (t_2 <= 400000.0)
		tmp = Float64(t_0 - Float64(x / B));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x * (-1.0 / tan(B));
	t_2 = t_0 + t_1;
	t_3 = t_1 + (1.0 / B);
	tmp = 0.0;
	if (t_2 <= -100000000000.0)
		tmp = t_3;
	elseif (t_2 <= 400000.0)
		tmp = t_0 - (x / B);
	else
		tmp = t_3;
	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[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -100000000000.0], t$95$3, If[LessEqual[t$95$2, 400000.0], N[(t$95$0 - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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))) < -1e11 or 4e5 < (+.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. Taylor expanded in B around 0

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

      1. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -1e11 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 4e5

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. un-div-inv N/A

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

      8. neg-mul-1 N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-sin.f64 N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq -100000000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{elif}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq 400000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -34000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 58000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -34000000000.0)
   (/ (- x) (tan B))
   (if (<= x 58000000.0) (- (/ 1.0 (sin B)) (/ x B)) (* x (/ -1.0 (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -34000000000.0) {
		tmp = -x / tan(B);
	} else if (x <= 58000000.0) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = x * (-1.0 / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-34000000000.0d0)) then
        tmp = -x / tan(b)
    else if (x <= 58000000.0d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = x * ((-1.0d0) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -34000000000.0) {
		tmp = -x / Math.tan(B);
	} else if (x <= 58000000.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = x * (-1.0 / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -34000000000.0:
		tmp = -x / math.tan(B)
	elif x <= 58000000.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = x * (-1.0 / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -34000000000.0)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (x <= 58000000.0)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -34000000000.0)
		tmp = -x / tan(B);
	elseif (x <= 58000000.0)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = x * (-1.0 / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -34000000000.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 58000000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e10

   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 inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   if -3.4e10 < x < 5.8e7

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. un-div-inv N/A

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

      8. neg-mul-1 N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-sin.f64 N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 97.8

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

   7. Applied rewrites 97.8%

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

   if 5.8e7 < 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 x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -34000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 58000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.75)
   (/ (- x) (tan B))
   (if (<= x 1.0) (/ 1.0 (sin B)) (* x (/ -1.0 (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = -x / tan(B);
	} else if (x <= 1.0) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = x * (-1.0 / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.75d0)) then
        tmp = -x / tan(b)
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = x * ((-1.0d0) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = -x / Math.tan(B);
	} else if (x <= 1.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = x * (-1.0 / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.75:
		tmp = -x / math.tan(B)
	elif x <= 1.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = x * (-1.0 / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.75)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (x <= 1.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.75)
		tmp = -x / tan(B);
	elseif (x <= 1.0)
		tmp = 1.0 / sin(B);
	else
		tmp = x * (-1.0 / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.75], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75

   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 inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.7%

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

   if -1.75 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 1 < 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 x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.75) 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 <= -1.75) {
		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 <= (-1.75d0)) 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 <= -1.75) {
		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 <= -1.75:
		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 <= -1.75)
		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 <= -1.75)
		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, -1.75], 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 < -1.75 or 1 < 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 x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.3%

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

   if -1.75 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 6: 62.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 1.04e-10) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 1.04e-10) {
		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 <= 1.04d-10) 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 <= 1.04e-10) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 1.04e-10:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 1.04e-10)
		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 <= 1.04e-10)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 1.04e-10], 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 < 1.04e-10

   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 64.2

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

   5. Applied rewrites 64.2%

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

   if 1.04e-10 < B

   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 53.4

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

   5. Applied rewrites 53.4%

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

Alternative 7: 51.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. un-div-inv N/A

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

   8. neg-mul-1 N/A

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

   9. lift-tan.f64 N/A

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

   10. tan-quot N/A

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

   11. lift-sin.f64 N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in B around 0

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

   1. lower-/.f64 72.1

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

7. Applied rewrites 72.1%

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

8. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 47.7

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

10. Applied rewrites 47.7%

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

Alternative 8: 51.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
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. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 47.5

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

5. Applied rewrites 47.5%

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

Alternative 9: 51.2% accurate, 7.5× speedup

Math

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

FPCore

(FPCore (B x) :precision binary64 (/ (/ (fma B (- x) B) B) B))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 47.4

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

5. Applied rewrites 47.4%

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

7. Applied rewrites 36.6%

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

9. Applied rewrites 47.5%

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

Alternative 10: 50.2% 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. 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 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.6%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 48.5

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.5%

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

Alternative 11: 51.1% accurate, 15.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 47.4

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

5. Applied rewrites 47.4%

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

Alternative 12: 27.0% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 47.4

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

5. Applied rewrites 47.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.7%

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

Reproduce

herbie shell --seed 2024231 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))