VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 12.2s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

(FPCore (B x) :precision binary64 (- (/ 1.0 (sin B)) (/ 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 (1.0 / sin(B)) - (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 = (1.0d0 / sin(b)) - (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 (1.0 / Math.sin(B)) - (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 (1.0 / math.sin(B)) - (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(Float64(1.0 / sin(B)) - 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 = (1.0 / sin(B)) - (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[(1.0 / N[Sin[B], $MachinePrecision]), $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))))): 28 points increase in error, 5 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. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	13248

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

Alternative 2
Error	1.0
Cost	7240

\[\begin{array}{l} t_0 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;x \leq -250950.48818132392:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0004292820693307226:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.4
Cost	6984

\[\begin{array}{l} t_0 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;x \leq -3.133766120598125 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.3854096834024 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.8
Cost	6920

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -250950.48818132392:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.5780207442592606:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.1
Cost	6856

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;B \leq -30524412289036350:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 0.2648088307626092:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	35.9
Cost	704

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

Alternative 7
Error	36.8
Cost	520

\[\begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -28548169.517780393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 18.821486917759:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	36.0
Cost	320

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

Alternative 9
Error	45.0
Cost	192

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

Reproduce

herbie shell --seed 2022297 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))