VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.9s

Alternatives: 11

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 99.6%

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ t_1 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -200000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2000:\\ \;\;\;\;\frac{x + -1}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (sin B)) (* x (/ -1.0 (tan B)))))
        (t_1 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= t_0 -200000000000.0)
     t_1
     (if (<= t_0 2000.0) (/ (+ x -1.0) (- (sin B))) t_1))))

C

double code(double B, double x) {
	double t_0 = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	double t_1 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (t_0 <= -200000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2000.0) {
		tmp = (x + -1.0) / -sin(B);
	} else {
		tmp = t_1;
	}
	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)) + (x * ((-1.0d0) / tan(b)))
    t_1 = (1.0d0 / b) - (x / tan(b))
    if (t_0 <= (-200000000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2000.0d0) then
        tmp = (x + (-1.0d0)) / -sin(b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 / Math.sin(B)) + (x * (-1.0 / Math.tan(B)));
	double t_1 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (t_0 <= -200000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2000.0) {
		tmp = (x + -1.0) / -Math.sin(B);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / math.sin(B)) + (x * (-1.0 / math.tan(B)))
	t_1 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if t_0 <= -200000000000.0:
		tmp = t_1
	elif t_0 <= 2000.0:
		tmp = (x + -1.0) / -math.sin(B)
	else:
		tmp = t_1
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(-1.0 / tan(B))))
	t_1 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -200000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2000.0)
		tmp = Float64(Float64(x + -1.0) / Float64(-sin(B)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	t_1 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (t_0 <= -200000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2000.0)
		tmp = (x + -1.0) / -sin(B);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000000000.0], t$95$1, If[LessEqual[t$95$0, 2000.0], N[(N[(x + -1.0), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

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

   1. Initial program 99.6%

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

      1. 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. neg-mul-1 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l/ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. un-div-inv N/A

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

      12. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 95.7

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

   11. Applied rewrites 95.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq -200000000000:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq 2000:\\ \;\;\;\;\frac{x + -1}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))