Average Error: 59.9 → 0.1
Time: 15.8s
Precision: binary64
Cost: 26688
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x} \]
↓
\[\frac{x}{\frac{\mathsf{fma}\left(0.0004938271604938272, {x}^{4}, 0.1111111111111111\right) + \left(x \cdot x\right) \cdot -0.007407407407407408}{\mathsf{fma}\left(1.0973936899862826 \cdot 10^{-5}, {x}^{6}, 0.037037037037037035\right)}} \]
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (/
  x
  (/
   (+
    (fma 0.0004938271604938272 (pow x 4.0) 0.1111111111111111)
    (* (* x x) -0.007407407407407408))
   (fma 1.0973936899862826e-5 (pow x 6.0) 0.037037037037037035))))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return x / ((fma(0.0004938271604938272, pow(x, 4.0), 0.1111111111111111) + ((x * x) * -0.007407407407407408)) / fma(1.0973936899862826e-5, pow(x, 6.0), 0.037037037037037035));
}

function code(x)
	return Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)))
end
↓
function code(x)
	return Float64(x / Float64(Float64(fma(0.0004938271604938272, (x ^ 4.0), 0.1111111111111111) + Float64(Float64(x * x) * -0.007407407407407408)) / fma(1.0973936899862826e-5, (x ^ 6.0), 0.037037037037037035)))
end

code[x_] := N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x / N[(N[(N[(0.0004938271604938272 * N[Power[x, 4.0], $MachinePrecision] + 0.1111111111111111), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * -0.007407407407407408), $MachinePrecision]), $MachinePrecision] / N[(1.0973936899862826e-5 * N[Power[x, 6.0], $MachinePrecision] + 0.037037037037037035), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\frac{\mathsf{fma}\left(0.0004938271604938272, {x}^{4}, 0.1111111111111111\right) + \left(x \cdot x\right) \cdot -0.007407407407407408}{\mathsf{fma}\left(1.0973936899862826 \cdot 10^{-5}, {x}^{6}, 0.037037037037037035\right)}}

Error

Target

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

Derivation

  1. Initial program 59.9

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

  2. Simplified59.9

    \[\leadsto \color{blue}{\frac{1}{x} + \frac{-1}{\tan x}} \]
    Proof
    (+.f64 (/.f64 1 x) (/.f64 -1 (tan.f64 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 x) (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) (tan.f64 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 x) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 (tan.f64 x))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 x) (/.f64 1 (tan.f64 x)))): 0 points increase in error, 0 points decrease in error

  3. Taylor expanded in x around 0 0.4

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

  4. Simplified0.4

    \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.022222222222222223, x \cdot x, 0.3333333333333333\right)} \]
    Proof
    (*.f64 x (fma.f64 1/45 (*.f64 x x) 1/3)): 0 points increase in error, 0 points decrease in error
    (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/45 (*.f64 x x)) 1/3))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (Rewrite=> +-commutative_binary64 (+.f64 1/3 (*.f64 1/45 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 1/3 x) (*.f64 (*.f64 1/45 (*.f64 x x)) x))): 1 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/3 x) (Rewrite<= associate-*r*_binary64 (*.f64 1/45 (*.f64 (*.f64 x x) x)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/3 x) (*.f64 1/45 (Rewrite<= unpow3_binary64 (pow.f64 x 3)))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr0.4

    \[\leadsto x \cdot \color{blue}{\frac{{\left(0.022222222222222223 \cdot \left(x \cdot x\right)\right)}^{3} + 0.037037037037037035}{\left(0.022222222222222223 \cdot \left(x \cdot x\right)\right) \cdot \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right) + \left(0.1111111111111111 - \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right) \cdot 0.3333333333333333\right)}} \]

  6. Applied egg-rr0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(0.0004938271604938272, {x}^{4}, 0.1111111111111111\right) - \left(x \cdot x\right) \cdot 0.007407407407407408}{\mathsf{fma}\left(1.0973936899862826 \cdot 10^{-5}, {x}^{6}, 0.037037037037037035\right)}}} \]

  7. Final simplification0.1

    \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(0.0004938271604938272, {x}^{4}, 0.1111111111111111\right) + \left(x \cdot x\right) \cdot -0.007407407407407408}{\mathsf{fma}\left(1.0973936899862826 \cdot 10^{-5}, {x}^{6}, 0.037037037037037035\right)}} \]

Alternatives

Alternative 1
Error0.4
Cost192
\[\frac{x}{3} \]

Error

Reproduce

herbie shell --seed 2022300 
(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))))