ENA, Section 1.4, Exercise 4a

Average Error: 30.2 → 0.3

Time: 21.2s

Precision: binary64

Cost: 26944

\[-1 \leq x \land x \leq 1\]

Math

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

↓

\[\mathsf{fma}\left(-0.0007275132275132275, {x}^{6}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right) + \left(\left(1 + -0.00023644179894179894 \cdot {x}^{8}\right) + -1\right)\right) \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (tan x)))

↓

(FPCore (x)
 :precision binary64
 (fma
  -0.0007275132275132275
  (pow x 6.0)
  (+
   (* (* x x) (fma (* x x) -0.06388888888888888 0.16666666666666666))
   (+ (+ 1.0 (* -0.00023644179894179894 (pow x 8.0))) -1.0))))

C

double code(double x) {
	return (x - sin(x)) / tan(x);
}

↓

double code(double x) {
	return fma(-0.0007275132275132275, pow(x, 6.0), (((x * x) * fma((x * x), -0.06388888888888888, 0.16666666666666666)) + ((1.0 + (-0.00023644179894179894 * pow(x, 8.0))) + -1.0)));
}

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / tan(x))
end

↓

function code(x)
	return fma(-0.0007275132275132275, (x ^ 6.0), Float64(Float64(Float64(x * x) * fma(Float64(x * x), -0.06388888888888888, 0.16666666666666666)) + Float64(Float64(1.0 + Float64(-0.00023644179894179894 * (x ^ 8.0))) + -1.0)))
end

Wolfram

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

↓

code[x_] := N[(-0.0007275132275132275 * N[Power[x, 6.0], $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.06388888888888888 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.00023644179894179894 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	30.2
Target	0.8
Herbie	0.3

\[0.16666666666666666 \cdot \left(x \cdot x\right) \]

Derivation

1. Initial program 30.2

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

2. Taylor expanded in x around 0 0.3

   \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2} + \left(-0.00023644179894179894 \cdot {x}^{8} + \left(-0.0007275132275132275 \cdot {x}^{6} + -0.06388888888888888 \cdot {x}^{4}\right)\right)} \]

3. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.0007275132275132275, {x}^{6}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right), -0.00023644179894179894 \cdot {x}^{8}\right)\right)} \]
   Proof
   (fma.f64 -11/15120 (pow.f64 x 6) (fma.f64 (*.f64 x x) (fma.f64 (*.f64 x x) -23/360 1/6) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (fma.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (fma.f64 (*.f64 x x) -23/360 1/6) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (fma.f64 (pow.f64 x 2) (fma.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) -23/360 1/6) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (fma.f64 (pow.f64 x 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 x 2) -23/360) 1/6)) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (fma.f64 (pow.f64 x 2) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -23/360 (pow.f64 x 2))) 1/6) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (fma.f64 (pow.f64 x 2) (Rewrite<= +-commutative_binary64 (+.f64 1/6 (*.f64 -23/360 (pow.f64 x 2)))) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 x 2) (+.f64 1/6 (*.f64 -23/360 (pow.f64 x 2)))) (*.f64 -143/604800 (pow.f64 x 8))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 1/6 (pow.f64 x 2)) (*.f64 (*.f64 -23/360 (pow.f64 x 2)) (pow.f64 x 2)))) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (+.f64 (+.f64 (*.f64 1/6 (pow.f64 x 2)) (Rewrite<= associate-*r*_binary64 (*.f64 -23/360 (*.f64 (pow.f64 x 2) (pow.f64 x 2))))) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (+.f64 (+.f64 (*.f64 1/6 (pow.f64 x 2)) (*.f64 -23/360 (Rewrite=> pow-sqr_binary64 (pow.f64 x (*.f64 2 2))))) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -11/15120 (pow.f64 x 6) (+.f64 (+.f64 (*.f64 1/6 (pow.f64 x 2)) (*.f64 -23/360 (pow.f64 x (Rewrite=> metadata-eval 4)))) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -11/15120 (pow.f64 x 6)) (+.f64 (+.f64 (*.f64 1/6 (pow.f64 x 2)) (*.f64 -23/360 (pow.f64 x 4))) (*.f64 -143/604800 (pow.f64 x 8))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -11/15120 (pow.f64 x 6)) (+.f64 (*.f64 1/6 (pow.f64 x 2)) (*.f64 -23/360 (pow.f64 x 4)))) (*.f64 -143/604800 (pow.f64 x 8)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 -11/15120 (pow.f64 x 6)) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -23/360 (pow.f64 x 4)) (*.f64 1/6 (pow.f64 x 2))))) (*.f64 -143/604800 (pow.f64 x 8))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -11/15120 (pow.f64 x 6)) (*.f64 -23/360 (pow.f64 x 4))) (*.f64 1/6 (pow.f64 x 2)))) (*.f64 -143/604800 (pow.f64 x 8))): 1 points increase in error, 0 points decrease in error
   (Rewrite=> associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -11/15120 (pow.f64 x 6)) (*.f64 -23/360 (pow.f64 x 4))) (+.f64 (*.f64 1/6 (pow.f64 x 2)) (*.f64 -143/604800 (pow.f64 x 8))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 1/6 (pow.f64 x 2)) (*.f64 -143/604800 (pow.f64 x 8))) (+.f64 (*.f64 -11/15120 (pow.f64 x 6)) (*.f64 -23/360 (pow.f64 x 4))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 1/6 (pow.f64 x 2)) (+.f64 (*.f64 -143/604800 (pow.f64 x 8)) (+.f64 (*.f64 -11/15120 (pow.f64 x 6)) (*.f64 -23/360 (pow.f64 x 4)))))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.3

   \[\leadsto \mathsf{fma}\left(-0.0007275132275132275, {x}^{6}, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right) + -0.00023644179894179894 \cdot {x}^{8}}\right) \]

5. Applied egg-rr 0.3

   \[\leadsto \mathsf{fma}\left(-0.0007275132275132275, {x}^{6}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right) + \color{blue}{\left(\left(1 + -0.00023644179894179894 \cdot {x}^{8}\right) - 1\right)}\right) \]

6. Final simplification 0.3

   \[\leadsto \mathsf{fma}\left(-0.0007275132275132275, {x}^{6}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right) + \left(\left(1 + -0.00023644179894179894 \cdot {x}^{8}\right) + -1\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	20672

\[\mathsf{fma}\left(-0.0007275132275132275, {x}^{6}, -0.00023644179894179894 \cdot {x}^{8} + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.06388888888888888\right) + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right) \]

Alternative 2
Error	0.3
Cost	19968

\[\mathsf{fma}\left(-0.0007275132275132275, {x}^{6}, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.06388888888888888, 0.16666666666666666\right)\right)\right) \]

Alternative 3
Error	0.4
Cost	6976

\[\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right) \]

Alternative 4
Error	0.8
Cost	320

\[x \cdot \left(x \cdot 0.16666666666666666\right) \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 4a"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :herbie-target
  (* 0.16666666666666666 (* x x))

  (/ (- x (sin x)) (tan x)))