bug323 (missed optimization)

Average Error: 59.6 → 57.3

Time: 8.3s

Precision: binary64

Cost: 71552

\[0 \leq x \land x \leq 0.5\]

Math

\[\cos^{-1} \left(1 - x\right) \]

↓

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{0.25 \cdot {\pi}^{2}}, -{t_0}^{2}\right)}{\pi \cdot 0.5 + t_0} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (fma
     (cbrt (pow (* PI 0.5) 4.0))
     (cbrt (* 0.25 (pow PI 2.0)))
     (- (pow t_0 2.0)))
    (+ (* PI 0.5) t_0))))

C

double code(double x) {
	return acos((1.0 - x));
}

↓

double code(double x) {
	double t_0 = asin((1.0 - x));
	return fma(cbrt(pow((((double) M_PI) * 0.5), 4.0)), cbrt((0.25 * pow(((double) M_PI), 2.0))), -pow(t_0, 2.0)) / ((((double) M_PI) * 0.5) + t_0);
}

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

↓

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(fma(cbrt((Float64(pi * 0.5) ^ 4.0)), cbrt(Float64(0.25 * (pi ^ 2.0))), Float64(-(t_0 ^ 2.0))) / Float64(Float64(pi * 0.5) + t_0))
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + (-N[Power[t$95$0, 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[(Pi * 0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Target

Original	59.6
Target	0.0
Herbie	57.3

\[2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \]

Derivation

1. Initial program 59.6

   \[\cos^{-1} \left(1 - x\right) \]

2. Applied egg-rr 59.6

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]

3. Applied egg-rr 57.3

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{0.25 \cdot {\pi}^{2}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

4. Final simplification 57.3

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{0.25 \cdot {\pi}^{2}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

Alternatives

Alternative 1
Error	57.3
Cost	45312

\[\mathsf{fma}\left(\sqrt{\pi} \cdot \sqrt{0.5}, \sqrt{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

Alternative 2
Error	57.4
Cost	20224

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ 0.3333333333333333 \cdot \left(t_0 \cdot -2\right) + \left(\pi \cdot 0.5 + t_0 \cdot -0.3333333333333333\right) \end{array} \]

Alternative 3
Error	57.4
Cost	20224

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \left(\pi \cdot 0.5 - 0.3333333333333333 \cdot \left(2 \cdot t_0\right)\right) + t_0 \cdot -0.3333333333333333 \end{array} \]

Alternative 4
Error	57.9
Cost	19780

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0 + -1\right|\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \log \left(e^{t_0}\right)\right)\\ \end{array} \]

Alternative 5
Error	57.9
Cost	13572

\[\begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 + 1\right) - \left(1 + \sin^{-1} \left(1 - x\right)\right)\\ \end{array} \]

Alternative 6
Error	57.9
Cost	13380

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0 + -1\right|\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \end{array} \]

Alternative 7
Error	57.9
Cost	13188

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\pi - t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \end{array} \]

Alternative 8
Error	59.6
Cost	6848

\[-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right) \]

Alternative 9
Error	59.6
Cost	6592

\[\cos^{-1} \left(1 - x\right) \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))