bug323 (missed optimization)

Average Accuracy: 6.8% → 10.4%

Time: 11.5s

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	6.8%
Target	100.0%
Herbie	10.4%

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

Derivation

1. Initial program 6.8%

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

2. Applied egg-rr 6.8%

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

   [Start] 6.8	\[ \cos^{-1} \left(1 - x\right) \]
   acos-asin [=>] 6.8	\[ \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]
   flip-- [=>] 6.8	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]
   div-inv [=>] 6.8	\[ \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   div-inv [=>] 6.8	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   div-inv [=>] 6.8	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

3. Applied egg-rr 10.4%

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

   [Start] 6.8	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   add-cube-cbrt [=>] 5.0	\[ \frac{\color{blue}{\left(\sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)} \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}\right) \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   fma-neg [=>] 5.0	\[ \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)} \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}, \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}, -\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

4. Final simplification 10.4%

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

Alternatives

Alternative 1
Accuracy	10.3%
Cost	52288

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{1 + \left(0.25 \cdot {\pi}^{2} - e^{\mathsf{log1p}\left({t_0}^{2}\right)}\right)}{\pi \cdot 0.5 + t_0} \end{array} \]

Alternative 2
Accuracy	10.3%
Cost	45696

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \mathsf{fma}\left(-t_1, t_1, t_0\right) + \cos^{-1} \left(1 - x\right) \end{array} \]

Alternative 3
Accuracy	10.3%
Cost	26688

\[\begin{array}{l} t_0 := \sqrt{2 + \cos^{-1} \left(1 - x\right)}\\ \left(t_0 + -1\right) \cdot \left(1 + t_0\right) + -1 \end{array} \]

Alternative 4
Accuracy	9.4%
Cost	19844

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt[3]{{t_0}^{3}}\\ \end{array} \]

Alternative 5
Accuracy	9.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 6
Accuracy	6.8%
Cost	13184

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

Alternative 7
Accuracy	6.8%
Cost	7232

\[-1 + \left(-1 + \left(2 + \left(-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right)\right)\right)\right) \]

Alternative 8
Accuracy	6.8%
Cost	6848

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

Alternative 9
Accuracy	6.8%
Cost	6848

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

Alternative 10
Accuracy	6.8%
Cost	6592

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

Reproduce

herbie shell --seed 2023138 
(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)))