Average Error: 0.2 → 0.2
Time: 11.2s
Precision: binary64
Cost: 13248
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
↓
\[\frac{1}{\sin B} - \frac{x}{\tan B} \]
(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))))
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));
}

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

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));
}

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))

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

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

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]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
    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 simplification0.2

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

Alternatives

Alternative 1
Error1.0
Cost13448
\[\begin{array}{l} \mathbf{if}\;x \leq -1.68:\\ \;\;\;\;\left(\frac{1}{B} + -1\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 115000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \]
Alternative 2
Error1.1
Cost7112
\[\begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error0.9
Cost7112
\[\begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error0.9
Cost7112
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.58:\\ \;\;\;\;\left(\frac{1}{B} + -1\right) - t_0\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]
Alternative 5
Error18.1
Cost6856
\[\begin{array}{l} t_0 := \frac{1 - x}{B} + B \cdot 0.16666666666666666\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error35.0
Cost584
\[\begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -21000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error34.5
Cost584
\[\begin{array}{l} t_0 := \frac{1 - x}{B}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{B} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error36.3
Cost520
\[\begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error34.3
Cost448
\[\frac{1 - x}{B} + -1 \]
Alternative 10
Error61.8
Cost192
\[B \cdot 0.16666666666666666 \]
Alternative 11
Error44.6
Cost192
\[\frac{1}{B} \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))