VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 11.3s

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 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
   Proof
   (-.f64 (/.f64 1 (sin.f64 B)) (/.f64 x (tan.f64 B))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (sin.f64 B)) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1)) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (sin.f64 B)) (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 1 (tan.f64 B))))): 32 points increase in error, 6 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 1 (sin.f64 B)) (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))) (/.f64 1 (sin.f64 B)))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.3
Cost	13448

\[\begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2496.6943410533304:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3157180201846665 \cdot 10^{-20}:\\ \;\;\;\;{\sin B}^{-1} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13248

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

Alternative 3
Error	0.2
Cost	13248

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

Alternative 4
Error	0.9
Cost	7240

\[\begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2496.6943410533304:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.1953353148812451:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.9
Cost	7112

\[\begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2496.6943410533304:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.1953353148812451:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2496.6943410533304:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3157180201846665 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.6
Cost	6856

\[\begin{array}{l} t_0 := \left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \mathbf{if}\;x \leq -9.863705822434112 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3157180201846665 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	17.0
Cost	6720

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

Alternative 9
Error	35.6
Cost	704

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

Alternative 10
Error	37.0
Cost	520

\[\begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -9.863705822434112 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.1953353148812451:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	35.8
Cost	320

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

Alternative 12
Error	44.7
Cost	192

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

Reproduce

herbie shell --seed 2022290 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))