VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 9.6s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. associate-/r/ N/A

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

   9. associate-*l/ N/A

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

   10. sub-div N/A

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

   11. remove-double-div N/A

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

   12. lift-/.f64 N/A

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

   13. div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. div-inv N/A

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

   17. lift-/.f64 N/A

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

   18. remove-double-div N/A

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

   19. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B} + t\_0\\ t_2 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1 (+ (* x (/ -1.0 (tan B))) t_0))
        (t_2 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= t_1 -50000000.0) t_2 (if (<= t_1 2000000.0) t_0 t_2))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = (x * (-1.0 / tan(B))) + t_0;
	double t_2 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = (x * ((-1.0d0) / tan(b))) + t_0
    t_2 = (1.0d0 / b) - (x / tan(b))
    if (t_1 <= (-50000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 2000000.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = (x * (-1.0 / Math.tan(B))) + t_0;
	double t_2 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = (x * (-1.0 / math.tan(B))) + t_0
	t_2 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if t_1 <= -50000000.0:
		tmp = t_2
	elif t_1 <= 2000000.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + t_0)
	t_2 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 2000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 1.0 / sin(B);
	t_1 = (x * (-1.0 / tan(B))) + t_0;
	t_2 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 2000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, 2000000.0], t$95$0, t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -5e7 or 2e6 < (+.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 99.6

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

   7. Applied rewrites 99.6%

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

   if -5e7 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 2e6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B} \leq -50000000:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B} \leq 2000000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))