bug323 (missed optimization)

Average Accuracy: 6.9% → 10.5%

Time: 12.4s

Precision: binary64

Cost: 97664

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

Math

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

↓

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \mathsf{fma}\left(\pi, 0.5, t_0\right)\\ \mathsf{fma}\left(\frac{\pi}{\sqrt{t_1}}, 0.25 \cdot \left(\pi \cdot \sqrt{\frac{1}{t_0 + \pi \cdot 0.5}}\right), \frac{-{t_0}^{2}}{t_1}\right) \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (fma PI 0.5 t_0)))
   (fma
    (/ PI (sqrt t_1))
    (* 0.25 (* PI (sqrt (/ 1.0 (+ t_0 (* PI 0.5))))))
    (/ (- (pow t_0 2.0)) t_1))))

C

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

↓

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = fma(((double) M_PI), 0.5, t_0);
	return fma((((double) M_PI) / sqrt(t_1)), (0.25 * (((double) M_PI) * sqrt((1.0 / (t_0 + (((double) M_PI) * 0.5)))))), (-pow(t_0, 2.0) / t_1));
}

Julia

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

↓

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = fma(pi, 0.5, t_0)
	return fma(Float64(pi / sqrt(t_1)), Float64(0.25 * Float64(pi * sqrt(Float64(1.0 / Float64(t_0 + Float64(pi * 0.5)))))), Float64(Float64(-(t_0 ^ 2.0)) / t_1))
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]}, Block[{t$95$1 = N[(Pi * 0.5 + t$95$0), $MachinePrecision]}, N[(N[(Pi / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(Pi * N[Sqrt[N[(1.0 / N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-N[Power[t$95$0, 2.0], $MachinePrecision]) / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Target

Original	6.9%
Target	100.0%
Herbie	10.5%

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

Derivation

1. Initial program 6.5%

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

2. Applied egg-rr 6.5%

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

   [Start] 6.5	\[ \cos^{-1} \left(1 - x\right) \]
   acos-asin [=>] 6.5	\[ \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]
   flip-- [=>] 6.5	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]
   div-inv [=>] 6.5	\[ \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   metadata-eval [=>] 6.5	\[ \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   div-inv [=>] 6.5	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   metadata-eval [=>] 6.5	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   div-inv [=>] 6.5	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]
   metadata-eval [=>] 6.5	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

3. Applied egg-rr 10.2%

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

   [Start] 6.5	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   div-sub [=>] 6.5	\[ \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} - \frac{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
   associate-*l* [=>] 6.5	\[ \frac{\color{blue}{\pi \cdot \left(0.5 \cdot \left(\pi \cdot 0.5\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} - \frac{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   add-sqr-sqrt [=>] 10.2	\[ \frac{\pi \cdot \left(0.5 \cdot \left(\pi \cdot 0.5\right)\right)}{\color{blue}{\sqrt{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \cdot \sqrt{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}}} - \frac{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   times-frac [=>] 10.2	\[ \color{blue}{\frac{\pi}{\sqrt{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \cdot \frac{0.5 \cdot \left(\pi \cdot 0.5\right)}{\sqrt{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}}} - \frac{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   fma-neg [=>] 10.2	\[ \color{blue}{\mathsf{fma}\left(\frac{\pi}{\sqrt{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}}, \frac{0.5 \cdot \left(\pi \cdot 0.5\right)}{\sqrt{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}}, -\frac{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}\right)} \]

4. Simplified 10.3%

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

   [Start] 10.2	\[ \mathsf{fma}\left(\frac{\pi}{\sqrt{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}, \frac{0.25 \cdot \pi}{\sqrt{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}, -\frac{{\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}\right) \]
   associate-/l* [=>] 10.2	\[ \mathsf{fma}\left(\frac{\pi}{\sqrt{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}, \color{blue}{\frac{0.25}{\frac{\sqrt{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}{\pi}}}, -\frac{{\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}\right) \]
   associate-/r/ [=>] 10.3	\[ \mathsf{fma}\left(\frac{\pi}{\sqrt{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}, \color{blue}{\frac{0.25}{\sqrt{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \cdot \pi}, -\frac{{\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}\right) \]
   distribute-neg-frac [=>] 10.3	\[ \mathsf{fma}\left(\frac{\pi}{\sqrt{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}, \frac{0.25}{\sqrt{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \cdot \pi, \color{blue}{\frac{-{\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}\right) \]

5. Taylor expanded in x around 0 10.3%

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

6. Final simplification 10.3%

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

Alternatives

Alternative 1
Accuracy	10.5%
Cost	45888

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

Alternative 2
Accuracy	10.4%
Cost	32512

\[\pi \cdot 0.5 - {\left({\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{0.16666666666666666}\right)}^{18} \]

Alternative 3
Accuracy	10.4%
Cost	26048

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

Alternative 4
Accuracy	10.4%
Cost	19712

\[-2 + {\left(\sqrt[3]{2 + \cos^{-1} \left(1 - x\right)}\right)}^{3} \]

Alternative 5
Accuracy	10.4%
Cost	19712

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

Alternative 6
Accuracy	6.9%
Cost	19456

\[{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3} \]

Alternative 7
Accuracy	6.9%
Cost	13184

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

Alternative 8
Accuracy	6.9%
Cost	6848

\[3 \cdot \left(\cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right) \]

Alternative 9
Accuracy	6.9%
Cost	6592

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

Reproduce

herbie shell --seed 2023157 
(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)))