bug323 (missed optimization)

Average Accuracy: 7.1% → 10.7%

Time: 8.6s

Precision: binary64

Cost: 39040

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

Math

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

↓

\[\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, {\left(\pi \cdot 0.5\right)}^{0.75}, -\sin^{-1} \left(1 - x\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma (pow (* PI 0.5) 0.25) (pow (* PI 0.5) 0.75) (- (asin (- 1.0 x)))))

C

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

↓

double code(double x) {
	return fma(pow((((double) M_PI) * 0.5), 0.25), pow((((double) M_PI) * 0.5), 0.75), -asin((1.0 - x)));
}

Julia

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

↓

function code(x)
	return fma((Float64(pi * 0.5) ^ 0.25), (Float64(pi * 0.5) ^ 0.75), Float64(-asin(Float64(1.0 - x))))
end

Wolfram

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

↓

code[x_] := N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 0.75], $MachinePrecision] + (-N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Target

Original	7.1%
Target	100.0%
Herbie	10.7%

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

Derivation

1. Initial program 7.1%

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

2. Applied egg-rr 7.1%

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

   [Start] 7.1	\[ \cos^{-1} \left(1 - x\right) \]
   acos-asin [=>] 7.1	\[ \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]
   sub-neg [=>] 7.1	\[ \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
   div-inv [=>] 7.1	\[ \color{blue}{\pi \cdot \frac{1}{2}} + \left(-\sin^{-1} \left(1 - x\right)\right) \]
   metadata-eval [=>] 7.1	\[ \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

3. Simplified 7.1%

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

   [Start] 7.1	\[ \pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right) \]
   sub-neg [<=] 7.1	\[ \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

4. Applied egg-rr 10.6%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)} \]
   Proof

   [Start] 7.1	\[ \pi \cdot 0.5 - \sin^{-1} \left(1 - x\right) \]
   asin-acos [=>] 7.1	\[ \pi \cdot 0.5 - \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(1 - x\right)\right)} \]
   div-inv [=>] 7.1	\[ \pi \cdot 0.5 - \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(1 - x\right)\right) \]
   metadata-eval [=>] 7.1	\[ \pi \cdot 0.5 - \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(1 - x\right)\right) \]
   sub-neg [=>] 7.1	\[ \pi \cdot 0.5 - \color{blue}{\left(\pi \cdot 0.5 + \left(-\cos^{-1} \left(1 - x\right)\right)\right)} \]
   add-sqr-sqrt [=>] 10.6	\[ \pi \cdot 0.5 - \left(\color{blue}{\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}} + \left(-\cos^{-1} \left(1 - x\right)\right)\right) \]
   fma-def [=>] 10.6	\[ \pi \cdot 0.5 - \color{blue}{\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)} \]

5. Applied egg-rr 10.7%

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

   [Start] 10.6	\[ \pi \cdot 0.5 - \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right) \]
   add-sqr-sqrt [=>] 7.1	\[ \color{blue}{\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}} - \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right) \]
   add-sqr-sqrt [=>] 10.6	\[ \color{blue}{\left(\sqrt{\sqrt{\pi \cdot 0.5}} \cdot \sqrt{\sqrt{\pi \cdot 0.5}}\right)} \cdot \sqrt{\pi \cdot 0.5} - \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right) \]
   associate-*l* [=>] 10.6	\[ \color{blue}{\sqrt{\sqrt{\pi \cdot 0.5}} \cdot \left(\sqrt{\sqrt{\pi \cdot 0.5}} \cdot \sqrt{\pi \cdot 0.5}\right)} - \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right) \]
   fma-neg [=>] 10.6	\[ \color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{\pi \cdot 0.5}}, \sqrt{\sqrt{\pi \cdot 0.5}} \cdot \sqrt{\pi \cdot 0.5}, -\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right)} \]
   pow1/2 [=>] 10.6	\[ \mathsf{fma}\left(\sqrt{\color{blue}{{\left(\pi \cdot 0.5\right)}^{0.5}}}, \sqrt{\sqrt{\pi \cdot 0.5}} \cdot \sqrt{\pi \cdot 0.5}, -\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right) \]
   sqrt-pow1 [=>] 10.6	\[ \mathsf{fma}\left(\color{blue}{{\left(\pi \cdot 0.5\right)}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{\pi \cdot 0.5}} \cdot \sqrt{\pi \cdot 0.5}, -\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right) \]
   metadata-eval [=>] 10.6	\[ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{\pi \cdot 0.5}} \cdot \sqrt{\pi \cdot 0.5}, -\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right) \]
   pow1/2 [=>] 10.6	\[ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, \sqrt{\color{blue}{{\left(\pi \cdot 0.5\right)}^{0.5}}} \cdot \sqrt{\pi \cdot 0.5}, -\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right) \]
   sqrt-pow1 [=>] 10.6	\[ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, \color{blue}{{\left(\pi \cdot 0.5\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \sqrt{\pi \cdot 0.5}, -\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right) \]
   metadata-eval [=>] 10.6	\[ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, {\left(\pi \cdot 0.5\right)}^{\color{blue}{0.25}} \cdot \sqrt{\pi \cdot 0.5}, -\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right) \]

6. Applied egg-rr 10.7%

   \[\leadsto \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, \color{blue}{{\left(\pi \cdot 0.5\right)}^{0.75}}, -\sin^{-1} \left(1 - x\right)\right) \]
   Proof

   [Start] 10.7	\[ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, {\left(\pi \cdot 0.5\right)}^{0.25} \cdot \sqrt{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \]
   pow1/2 [=>] 10.7	\[ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, {\left(\pi \cdot 0.5\right)}^{0.25} \cdot \color{blue}{{\left(\pi \cdot 0.5\right)}^{0.5}}, -\sin^{-1} \left(1 - x\right)\right) \]
   pow-prod-up [=>] 10.7	\[ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, \color{blue}{{\left(\pi \cdot 0.5\right)}^{\left(0.25 + 0.5\right)}}, -\sin^{-1} \left(1 - x\right)\right) \]
   metadata-eval [=>] 10.7	\[ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, {\left(\pi \cdot 0.5\right)}^{\color{blue}{0.75}}, -\sin^{-1} \left(1 - x\right)\right) \]

7. Final simplification 10.7%

   \[\leadsto \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.25}, {\left(\pi \cdot 0.5\right)}^{0.75}, -\sin^{-1} \left(1 - x\right)\right) \]

Alternatives

Alternative 1
Accuracy	9.7%
Cost	26372

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1 + \sqrt{{\left(t_0 + -1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{-1 + \left(1 + t_0\right)}\right)\\ \end{array} \]

Alternative 2
Accuracy	9.7%
Cost	26372

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := t_0 + -1\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1 + \sqrt{{t_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + {\left(\sqrt[3]{t_1}\right)}^{3}\\ \end{array} \]

Alternative 3
Accuracy	9.7%
Cost	26308

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{-1 + \left(1 + t_0\right)}\right)\\ \end{array} \]

Alternative 4
Accuracy	9.7%
Cost	26180

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({t_0}^{0.3333333333333333}\right)}^{3}\\ \end{array} \]

Alternative 5
Accuracy	10.6%
Cost	26048

\[\pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]

Alternative 6
Accuracy	10.6%
Cost	26048

\[\pi \cdot 0.5 - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \]

Alternative 7
Accuracy	9.7%
Cost	19844

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \end{array} \]

Alternative 8
Accuracy	7.1%
Cost	6848

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

Alternative 9
Accuracy	7.1%
Cost	6848

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

Alternative 10
Accuracy	7.1%
Cost	6592

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

Reproduce

herbie shell --seed 2023147 
(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)))