VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 10.0s

Precision: binary64

Cost: 19776

Math

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

↓

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

FPCore

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

↓

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

↓

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

↓

public static double code(double B, double x) {
	return (1.0 / Math.sin(B)) - ((Math.cos(B) * x) / Math.sin(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)) - ((math.cos(B) * x) / math.sin(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(Float64(cos(B) * x) / sin(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)) - ((cos(B) * x) / 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]

↓

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

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

   [Start] 0.2	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ \color{blue}{\frac{1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-97 [=>] 0.2	\[ \frac{1}{\sin B} + \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-108 [=>] 0.2	\[ \color{blue}{\left(0 + \frac{1}{\sin B}\right) - x \cdot \frac{1}{\tan B}} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ \color{blue}{\left(\frac{1}{\sin B} + 0\right)} - x \cdot \frac{1}{\tan B} \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 0.2	\[ \color{blue}{\frac{1}{\sin B}} - x \cdot \frac{1}{\tan B} \]

3. Taylor expanded in x around 0 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	19776

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

Alternative 2
Error	2.1
Cost	13512

\[\begin{array}{l} t_0 := -\frac{\cos B \cdot x}{\sin B}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.2
Cost	13448

\[\begin{array}{l} t_0 := -\frac{\cos B \cdot x}{\sin B}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.2
Cost	13376

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

Alternative 5
Error	10.0
Cost	7496

\[\begin{array}{l} t_0 := \left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6000000000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	18.0
Cost	6920

\[\begin{array}{l} t_0 := -\frac{x}{\sin B}\\ \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	18.6
Cost	6856

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;B \leq -5.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 0.0132:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	18.5
Cost	6848

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

Alternative 9
Error	36.8
Cost	520

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	35.6
Cost	448

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

Alternative 11
Error	35.6
Cost	320

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

Alternative 12
Error	45.1
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))