VandenBroeck and Keller, Equation (24)

Average Error: 0.33% → 0.23%

Time: 12.3s

Precision: binary64

Cost: 19584

Math

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

↓

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

FPCore

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

↓

(FPCore (B x) :precision binary64 (- (pow (sin B) -1.0) (/ 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 pow(sin(B), -1.0) - (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 = (sin(b) ** (-1.0d0)) - (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 Math.pow(Math.sin(B), -1.0) - (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 math.pow(math.sin(B), -1.0) - (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((sin(B) ^ -1.0) - 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 = (sin(B) ^ -1.0) - (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[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.33

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

2. Simplified 0.23

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

   [Start] 0.33	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   +-commutative [=>] 0.33	\[ \color{blue}{\frac{1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
   unsub-neg [=>] 0.33	\[ \color{blue}{\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}} \]
   associate-*r/ [=>] 0.23	\[ \frac{1}{\sin B} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
   *-rgt-identity [=>] 0.23	\[ \frac{1}{\sin B} - \frac{\color{blue}{x}}{\tan B} \]

3. Applied egg-rr 0.23

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

4. Final simplification 0.23

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

Alternatives

Alternative 1
Error	1.62%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;x \leq -520:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;{\sin B}^{-1} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]

Alternative 2
Error	0.23%
Cost	13248

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

Alternative 3
Error	1.62%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -520:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]

Alternative 4
Error	1.68%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-6} \lor \neg \left(x \leq 5.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5
Error	1.68%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]

Alternative 6
Error	2.46%
Cost	6921

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

Alternative 7
Error	29.21%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{B} - \frac{x}{B}\right) + B \cdot 0.16666666666666666\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \end{array} \]

Alternative 8
Error	55.97%
Cost	704

\[\frac{1 - x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right) \]

Alternative 9
Error	55.67%
Cost	704

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

Alternative 10
Error	55.66%
Cost	576

\[B \cdot 0.16666666666666666 + \frac{1 - x}{B} \]

Alternative 11
Error	56.95%
Cost	521

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

Alternative 12
Error	55.85%
Cost	448

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

Alternative 13
Error	55.85%
Cost	320

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

Alternative 14
Error	96.64%
Cost	192

\[B \cdot 0.16666666666666666 \]

Alternative 15
Error	70.05%
Cost	192

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

Reproduce

herbie shell --seed 2023121 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))