VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 10.9s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

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

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)) - ((1.0 + (x / tan(B))) + -1.0);
}

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)) - ((1.0d0 + (x / tan(b))) + (-1.0d0))
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)) - ((1.0 + (x / Math.tan(B))) + -1.0);
}

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)) - ((1.0 + (x / math.tan(B))) + -1.0)

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(Float64(1.0 + Float64(x / tan(B))) + -1.0))
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)) - ((1.0 + (x / tan(B))) + -1.0);
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[(N[(1.0 + N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.1
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -39000000000 \lor \neg \left(x \leq 3800000\right):\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{1}{\frac{B}{x}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13248

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

Alternative 3
Error	8.8
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -780000000000 \lor \neg \left(x \leq 4.5 \cdot 10^{+14}\right):\\ \;\;\;\;\left(B \cdot 0.16666666666666666 + \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{1}{\frac{B}{x}}\\ \end{array} \]

Alternative 4
Error	17.7
Cost	6857

\[\begin{array}{l} \mathbf{if}\;B \leq -0.03 \lor \neg \left(B \leq 0.031\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(B \cdot 0.16666666666666666 + \frac{1}{B}\right) + \left(B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\right)\\ \end{array} \]

Alternative 5
Error	17.7
Cost	6848

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

Alternative 6
Error	35.1
Cost	704

\[\left(B \cdot 0.16666666666666666 + \frac{1}{B}\right) - \frac{x}{B} \]

Alternative 7
Error	36.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \]

Alternative 8
Error	35.2
Cost	320

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

Alternative 9
Error	53.1
Cost	256

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

Alternative 10
Error	61.8
Cost	192

\[B \cdot 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))