VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 12.5s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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 = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

↓

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 -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

↓

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

   [Start] 0.2	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   +-commutative [=>] 0.2	\[ \color{blue}{\frac{1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
   unsub-neg [=>] 0.2	\[ \color{blue}{\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}} \]
   associate-*r/ [=>] 0.2	\[ \frac{1}{\sin B} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
   *-rgt-identity [=>] 0.2	\[ \frac{1}{\sin B} - \frac{\color{blue}{x}}{\tan B} \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.0
Cost	13384

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.66:\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{elif}\;x \leq 95000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\sin B - t_0\\ \end{array} \]

Alternative 2
Error	1.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-7} \lor \neg \left(x \leq 0.00024\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 0.054\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 4
Error	1.6
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.08 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;1 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5
Error	18.7
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-6} \lor \neg \left(x \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6
Error	32.7
Cost	6728

\[\begin{array}{l} \mathbf{if}\;B \leq -1.5:\\ \;\;\;\;\sin B\\ \mathbf{elif}\;B \leq 850:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} - \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin B\\ \end{array} \]

Alternative 7
Error	34.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;B \leq -14000:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} - \frac{x}{B}\right)\\ \end{array} \]

Alternative 8
Error	34.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;B \leq -14000:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \end{array} \]

Alternative 9
Error	36.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -7200000000 \lor \neg \left(x \leq 0.98\right):\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 10
Error	34.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;B \leq -14000:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]

Alternative 11
Error	36.4
Cost	521

\[\begin{array}{l} \mathbf{if}\;x \leq -7200000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 12
Error	34.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;B \leq -14000:\\ \;\;\;\;1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 13
Error	44.8
Cost	192

\[\frac{1}{B} \]

Reproduce

herbie shell --seed 2023066 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))