Average Error: 0.1 → 0.1
Time: 2.0s
Precision: binary64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
↓
\[3 + \mathsf{fma}\left(9, x \cdot x, x \cdot -12\right) \]
(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))
↓
(FPCore (x) :precision binary64 (+ 3.0 (fma 9.0 (* x x) (* x -12.0))))
double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}
↓
double code(double x) {
	return 3.0 + fma(9.0, (x * x), (x * -12.0));
}

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end
↓
function code(x)
	return Float64(3.0 + fma(9.0, Float64(x * x), Float64(x * -12.0)))
end

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(3.0 + N[(9.0 * N[(x * x), $MachinePrecision] + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
3 + \mathsf{fma}\left(9, x \cdot x, x \cdot -12\right)

Error

Bits error versus x

Target

Original0.1
Target0.1
Herbie0.1
\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right) \]

Derivation

  1. Initial program 0.1

    \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)} \]

  3. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{\left(3 + 9 \cdot {x}^{2}\right) - 12 \cdot x} \]

  4. Applied egg-rr0.1

    \[\leadsto \color{blue}{3 + \mathsf{fma}\left(9, x \cdot x, x \cdot -12\right)} \]

  5. Final simplification0.1

    \[\leadsto 3 + \mathsf{fma}\left(9, x \cdot x, x \cdot -12\right) \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))