VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 12.7s

Precision: binary64

Cost: 13440

Math

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

↓

\[\frac{x - \frac{x}{x \cdot \cos B}}{-\tan B} \]

FPCore

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

↓

(FPCore (B x) :precision binary64 (/ (- x (/ x (* x (cos B)))) (- (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 (x - (x / (x * cos(B)))) / -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 = (x - (x / (x * cos(b)))) / -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 (x - (x / (x * Math.cos(B)))) / -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 (x - (x / (x * math.cos(B)))) / -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(x - Float64(x / Float64(x * cos(B)))) / Float64(-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 = (x - (x / (x * cos(B)))) / -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[(x - N[(x / N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
   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/ [=>] 0.1	\[ \frac{1}{\sin B} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
   *-rgt-identity [=>] 0.1	\[ \frac{1}{\sin B} - \frac{\color{blue}{x}}{\tan B} \]

3. Applied egg-rr 9.0

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

4. Simplified 8.9

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

   [Start] 9.0	\[ \frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}} \]
   associate-/r* [=>] 0.3	\[ \color{blue}{\frac{\frac{\frac{\tan B}{x} - \sin B}{\sin B}}{\frac{\tan B}{x}}} \]
   div-sub [=>] 0.3	\[ \frac{\color{blue}{\frac{\frac{\tan B}{x}}{\sin B} - \frac{\sin B}{\sin B}}}{\frac{\tan B}{x}} \]
   *-inverses [=>] 0.3	\[ \frac{\frac{\frac{\tan B}{x}}{\sin B} - \color{blue}{1}}{\frac{\tan B}{x}} \]
   sub-neg [=>] 0.3	\[ \frac{\color{blue}{\frac{\frac{\tan B}{x}}{\sin B} + \left(-1\right)}}{\frac{\tan B}{x}} \]
   associate-/l/ [=>] 8.9	\[ \frac{\color{blue}{\frac{\tan B}{\sin B \cdot x}} + \left(-1\right)}{\frac{\tan B}{x}} \]
   *-commutative [<=] 8.9	\[ \frac{\frac{\tan B}{\color{blue}{x \cdot \sin B}} + \left(-1\right)}{\frac{\tan B}{x}} \]
   metadata-eval [=>] 8.9	\[ \frac{\frac{\tan B}{x \cdot \sin B} + \color{blue}{-1}}{\frac{\tan B}{x}} \]

5. Taylor expanded in B around inf 0.4

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

6. Applied egg-rr 0.3

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

7. Simplified 0.2

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

   [Start] 0.3	\[ \left(-\frac{x}{\tan B}\right) + \frac{1}{\cos B \cdot x} \cdot \frac{x}{\tan B} \]
   +-commutative [<=] 0.3	\[ \color{blue}{\frac{1}{\cos B \cdot x} \cdot \frac{x}{\tan B} + \left(-\frac{x}{\tan B}\right)} \]
   neg-mul-1 [=>] 0.3	\[ \frac{1}{\cos B \cdot x} \cdot \frac{x}{\tan B} + \color{blue}{-1 \cdot \frac{x}{\tan B}} \]
   distribute-rgt-in [<=] 0.3	\[ \color{blue}{\frac{x}{\tan B} \cdot \left(\frac{1}{\cos B \cdot x} + -1\right)} \]
   *-commutative [<=] 0.3	\[ \color{blue}{\left(\frac{1}{\cos B \cdot x} + -1\right) \cdot \frac{x}{\tan B}} \]
   associate-*r/ [=>] 0.3	\[ \color{blue}{\frac{\left(\frac{1}{\cos B \cdot x} + -1\right) \cdot x}{\tan B}} \]
   associate-*l/ [<=] 9.4	\[ \color{blue}{\frac{\frac{1}{\cos B \cdot x} + -1}{\tan B} \cdot x} \]
   remove-double-neg [<=] 9.4	\[ \frac{\frac{1}{\cos B \cdot x} + -1}{\tan B} \cdot \color{blue}{\left(-\left(-x\right)\right)} \]
   distribute-rgt-neg-in [<=] 9.4	\[ \color{blue}{-\frac{\frac{1}{\cos B \cdot x} + -1}{\tan B} \cdot \left(-x\right)} \]
   associate-/r/ [<=] 0.4	\[ -\color{blue}{\frac{\frac{1}{\cos B \cdot x} + -1}{\frac{\tan B}{-x}}} \]
   neg-mul-1 [=>] 0.4	\[ \color{blue}{-1 \cdot \frac{\frac{1}{\cos B \cdot x} + -1}{\frac{\tan B}{-x}}} \]
   associate-/r/ [=>] 9.4	\[ -1 \cdot \color{blue}{\left(\frac{\frac{1}{\cos B \cdot x} + -1}{\tan B} \cdot \left(-x\right)\right)} \]
   metadata-eval [<=] 9.4	\[ \color{blue}{\frac{1}{-1}} \cdot \left(\frac{\frac{1}{\cos B \cdot x} + -1}{\tan B} \cdot \left(-x\right)\right) \]
   associate-*l/ [=>] 0.3	\[ \frac{1}{-1} \cdot \color{blue}{\frac{\left(\frac{1}{\cos B \cdot x} + -1\right) \cdot \left(-x\right)}{\tan B}} \]
   times-frac [<=] 0.3	\[ \color{blue}{\frac{1 \cdot \left(\left(\frac{1}{\cos B \cdot x} + -1\right) \cdot \left(-x\right)\right)}{-1 \cdot \tan B}} \]

8. Final simplification 0.2

   \[\leadsto \frac{x - \frac{x}{x \cdot \cos B}}{-\tan B} \]

Alternatives

Alternative 1
Error	0.1
Cost	13248

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

Alternative 2
Error	0.2
Cost	13248

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

Alternative 3
Error	1.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 1.82 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 4
Error	1.8
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5
Error	1.2
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-6} \lor \neg \left(x \leq 1.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6
Error	18.1
Cost	6857

\[\begin{array}{l} \mathbf{if}\;B \leq -0.026 \lor \neg \left(B \leq 0.066\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 7
Error	36.4
Cost	521

\[\begin{array}{l} \mathbf{if}\;x \leq -1460000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 8
Error	35.5
Cost	320

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

Alternative 9
Error	61.9
Cost	192

\[B \cdot 0.16666666666666666 \]

Alternative 10
Error	44.7
Cost	192

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

Reproduce

herbie shell --seed 2023025 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))