Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Average Error: 0.3 → 0.2

Time: 2.2s

Precision: binary64

Math

\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-x \cdot x, 9, \left(x \cdot x\right) \cdot 9\right)\\ \mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot -9\right) + \left(t_0 + t_0\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (- (* x x)) 9.0 (* (* x x) 9.0))))
   (+ (fma x 6.0 (* (* x x) -9.0)) (+ t_0 t_0))))

C

double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

↓

double code(double x) {
	double t_0 = fma(-(x * x), 9.0, ((x * x) * 9.0));
	return fma(x, 6.0, ((x * x) * -9.0)) + (t_0 + t_0);
}

Julia

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end

↓

function code(x)
	t_0 = fma(Float64(-Float64(x * x)), 9.0, Float64(Float64(x * x) * 9.0))
	return Float64(fma(x, 6.0, Float64(Float64(x * x) * -9.0)) + Float64(t_0 + t_0))
end

Wolfram

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[((-N[(x * x), $MachinePrecision]) * 9.0 + N[(N[(x * x), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(x * 6.0 + N[(N[(x * x), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Target

Original	0.3
Target	0.2
Herbie	0.2

\[6 \cdot x - 9 \cdot \left(x \cdot x\right) \]

Derivation

1. Initial program 0.3

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]

2. Simplified 0.2

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -9, 6\right)} \]

3. Taylor expanded in x around 0 0.2

   \[\leadsto \color{blue}{6 \cdot x - 9 \cdot {x}^{2}} \]

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot -9\right) + \left(\mathsf{fma}\left(-x \cdot x, 9, 9 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(-x \cdot x, 9, 9 \cdot \left(x \cdot x\right)\right)\right)} \]

5. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot -9\right) + \left(\mathsf{fma}\left(-x \cdot x, 9, \left(x \cdot x\right) \cdot 9\right) + \mathsf{fma}\left(-x \cdot x, 9, \left(x \cdot x\right) \cdot 9\right)\right) \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6.0 x) (* 9.0 (* x x)))

  (* (* 3.0 (- 2.0 (* x 3.0))) x))