bug323 (missed optimization)

Average Error: 59.6 → 57.4

Time: 9.8s

Precision: binary64

Cost: 26048

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (sqrt (asin (- 1.0 x))) 2.0)))

C

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

↓

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(sqrt(asin((1.0 - x))), 2.0);
}

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

↓

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.sqrt(Math.asin((1.0 - x))), 2.0);
}

Python

def code(x):
	return math.acos((1.0 - x))

↓

def code(x):
	return (math.pi * 0.5) - math.pow(math.sqrt(math.asin((1.0 - x))), 2.0)

Julia

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

↓

function code(x)
	return Float64(Float64(pi * 0.5) - (sqrt(asin(Float64(1.0 - x))) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

↓

function tmp = code(x)
	tmp = (pi * 0.5) - (sqrt(asin((1.0 - x))) ^ 2.0);
end

Wolfram

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

↓

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Target

Original	59.6
Target	0.0
Herbie	57.4

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

Derivation

1. Initial program 59.6

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

2. Applied egg-rr 59.6

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

3. Simplified 59.6

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

   [Start] 59.6	\[ \pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right) \]
   sub-neg [<=] 59.6	\[ \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

4. Applied egg-rr 57.4

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

5. Final simplification 57.4

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

Alternatives

Alternative 1
Error	57.4
Cost	26048

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

Alternative 2
Error	57.4
Cost	19840

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

Alternative 3
Error	57.4
Cost	19840

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

Alternative 4
Error	59.6
Cost	13440

\[3 \cdot \left(\pi \cdot 0.16666666666666666 - \sin^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right) \]

Alternative 5
Error	59.6
Cost	13184

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

Alternative 6
Error	59.6
Cost	6848

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

Alternative 7
Error	59.6
Cost	6592

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

Reproduce

herbie shell --seed 2023060 
(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)))