VandenBroeck and Keller, Equation (24)

Average Accuracy: 99.7% → 99.8%

Time: 13.8s

Precision: binary64

Cost: 19648

Math

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

↓

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

FPCore

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

↓

(FPCore (B x) :precision binary64 (/ (- (/ (tan B) (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 ((tan(B) / 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 = ((tan(b) / 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 ((Math.tan(B) / 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 ((math.tan(B) / 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(Float64(tan(B) / sin(B)) - 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 = ((tan(B) / 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[(N[(N[Tan[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
   Step-by-step derivation

   [Start] 99.6	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   +-commutative [=>] 99.6	\[ \color{blue}{\frac{1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
   unsub-neg [=>] 99.6	\[ \color{blue}{\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}} \]
   associate-*r/ [=>] 99.7	\[ \frac{1}{\sin B} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
   *-rgt-identity [=>] 99.7	\[ \frac{1}{\sin B} - \frac{\color{blue}{x}}{\tan B} \]

3. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\tan B - x \cdot \sin B}{\sin B}}{\tan B}} \]
   Step-by-step derivation

   [Start] 99.7	\[ \frac{1}{\sin B} - \frac{x}{\tan B} \]
   frac-sub [=>] 87.4	\[ \color{blue}{\frac{1 \cdot \tan B - \sin B \cdot x}{\sin B \cdot \tan B}} \]
   associate-/r* [=>] 99.7	\[ \color{blue}{\frac{\frac{1 \cdot \tan B - \sin B \cdot x}{\sin B}}{\tan B}} \]
   *-un-lft-identity [<=] 99.7	\[ \frac{\frac{\color{blue}{\tan B} - \sin B \cdot x}{\sin B}}{\tan B} \]
   *-commutative [=>] 99.7	\[ \frac{\frac{\tan B - \color{blue}{x \cdot \sin B}}{\sin B}}{\tan B} \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{\color{blue}{\frac{\tan B}{\sin B} + \left(-\frac{\sin B}{\frac{\sin B}{x}}\right)}}{\tan B} \]
   Step-by-step derivation

   [Start] 99.7	\[ \frac{\frac{\tan B - x \cdot \sin B}{\sin B}}{\tan B} \]
   div-sub [=>] 99.7	\[ \frac{\color{blue}{\frac{\tan B}{\sin B} - \frac{x \cdot \sin B}{\sin B}}}{\tan B} \]
   sub-neg [=>] 99.7	\[ \frac{\color{blue}{\frac{\tan B}{\sin B} + \left(-\frac{x \cdot \sin B}{\sin B}\right)}}{\tan B} \]
   *-commutative [=>] 99.7	\[ \frac{\frac{\tan B}{\sin B} + \left(-\frac{\color{blue}{\sin B \cdot x}}{\sin B}\right)}{\tan B} \]
   associate-/l* [=>] 99.7	\[ \frac{\frac{\tan B}{\sin B} + \left(-\color{blue}{\frac{\sin B}{\frac{\sin B}{x}}}\right)}{\tan B} \]

5. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\frac{\tan B}{\sin B} - x}}{\tan B} \]
   Step-by-step derivation

   [Start] 99.7	\[ \frac{\frac{\tan B}{\sin B} + \left(-\frac{\sin B}{\frac{\sin B}{x}}\right)}{\tan B} \]
   sub-neg [<=] 99.7	\[ \frac{\color{blue}{\frac{\tan B}{\sin B} - \frac{\sin B}{\frac{\sin B}{x}}}}{\tan B} \]
   associate-/r/ [=>] 99.7	\[ \frac{\frac{\tan B}{\sin B} - \color{blue}{\frac{\sin B}{\sin B} \cdot x}}{\tan B} \]
   *-inverses [=>] 99.7	\[ \frac{\frac{\tan B}{\sin B} - \color{blue}{1} \cdot x}{\tan B} \]
   *-lft-identity [=>] 99.7	\[ \frac{\frac{\tan B}{\sin B} - \color{blue}{x}}{\tan B} \]

6. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13248

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

Alternative 2
Accuracy	98.6%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -2.85 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;\left(\frac{1}{B} + -1\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1100000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]

Alternative 4
Accuracy	98.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-7} \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5
Accuracy	97.6%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6
Accuracy	75.1%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 7
Accuracy	49.6%
Cost	521

\[\begin{array}{l} \mathbf{if}\;x \leq -30000000000000 \lor \neg \left(x \leq 2.3 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 8
Accuracy	51.5%
Cost	320

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

Alternative 9
Accuracy	26.4%
Cost	192

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

Reproduce

herbie shell --seed 2023164 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))