Average Error: 59.9 → 0.4
Time: 12.6s
Precision: binary64
Cost: 192
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x} \]
↓
\[\frac{x}{3} \]
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x) :precision binary64 (/ x 3.0))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return x / 3.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) - (1.0d0 / tan(x))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = x / 3.0d0
end function

public static double code(double x) {
	return (1.0 / x) - (1.0 / Math.tan(x));
}
↓
public static double code(double x) {
	return x / 3.0;
}

def code(x):
	return (1.0 / x) - (1.0 / math.tan(x))
↓
def code(x):
	return x / 3.0

function code(x)
	return Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)))
end
↓
function code(x)
	return Float64(x / 3.0)
end

function tmp = code(x)
	tmp = (1.0 / x) - (1.0 / tan(x));
end
↓
function tmp = code(x)
	tmp = x / 3.0;
end

code[x_] := N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x / 3.0), $MachinePrecision]

\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{3}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \]

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded in x around 0 0.4

    \[\leadsto \color{blue}{0.3333333333333333 \cdot x + 0.022222222222222223 \cdot {x}^{3}} \]

  3. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 0.022222222222222223 \cdot {x}^{3}\right)} \]
    Proof
    (fma.f64 1/3 x (*.f64 1/45 (pow.f64 x 3))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3)))): 1 points increase in error, 0 points decrease in error

  4. Applied egg-rr29.5

    \[\leadsto \color{blue}{\frac{\left(0.3333333333333333 \cdot x\right) \cdot \left(0.3333333333333333 \cdot x\right) - {x}^{6} \cdot 0.0004938271604938272}{0.3333333333333333 \cdot x - 0.022222222222222223 \cdot {x}^{3}}} \]

  5. Simplified29.5

    \[\leadsto \color{blue}{\frac{0.1111111111111111 \cdot \left(x \cdot x\right) - {x}^{6} \cdot 0.0004938271604938272}{0.3333333333333333 \cdot x - 0.022222222222222223 \cdot {x}^{3}}} \]
    Proof
    (/.f64 (-.f64 (*.f64 1/9 (*.f64 x x)) (*.f64 (pow.f64 x 6) 1/2025)) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 1/3 1/3)) (*.f64 x x)) (*.f64 (pow.f64 x 6) 1/2025)) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 1/3 x) (*.f64 1/3 x))) (*.f64 (pow.f64 x 6) 1/2025)) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3)))): 25 points increase in error, 25 points decrease in error
    (/.f64 (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (*.f64 1/3 x) (*.f64 1/3 x)) 0)) (*.f64 (pow.f64 x 6) 1/2025)) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3)))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (+.f64 (*.f64 (*.f64 1/3 x) (*.f64 1/3 x)) 0) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3)))) (/.f64 (*.f64 (pow.f64 x 6) 1/2025) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3)))))): 1 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 (Rewrite=> +-rgt-identity_binary64 (*.f64 (*.f64 1/3 x) (*.f64 1/3 x))) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3)))) (/.f64 (*.f64 (pow.f64 x 6) 1/2025) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 (*.f64 1/3 x) (*.f64 1/3 x)) (*.f64 (pow.f64 x 6) 1/2025)) (-.f64 (*.f64 1/3 x) (*.f64 1/45 (pow.f64 x 3))))): 0 points increase in error, 1 points decrease in error

  6. Taylor expanded in x around 0 0.7

    \[\leadsto \color{blue}{0.3333333333333333 \cdot x} \]

  7. Simplified0.4

    \[\leadsto \color{blue}{\frac{x}{3}} \]
    Proof
    (/.f64 x 3): 0 points increase in error, 0 points decrease in error
    (/.f64 x (Rewrite<= metadata-eval (/.f64 1 1/3))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 1/3) 1)): 75 points increase in error, 0 points decrease in error
    (Rewrite=> /-rgt-identity_binary64 (*.f64 x 1/3)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-commutative_binary64 (*.f64 1/3 x)): 0 points increase in error, 0 points decrease in error

  8. Final simplification0.4

    \[\leadsto \frac{x}{3} \]

Alternatives

Alternative 1
Error0.7
Cost192
\[x \cdot 0.3333333333333333 \]

Error

Reproduce

herbie shell --seed 2022328 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0))) (- (/ 1.0 x) (/ 1.0 (tan x))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))